Your Car Is Skidding to a Stop From a High Speed You May Want to Review (Pages 112 - 114)
Key Terms
- banked curve
- curve in a road that is sloping in a fashion that helps a vehicle negotiate the bend
- centripetal strength
- whatever net force causing uniform circular motility
- Coriolis strength
- inertial force causing the apparent deflection of moving objects when viewed in a rotating frame of reference
- drag forcefulness
- force that e'er opposes the movement of an object in a fluid; unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid
- friction
- force that opposes relative move or attempts at motion between systems in contact
- ideal banking
- sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the cyberspace external force on the vehicle equals the horizontal centripetal force in the absence of friction
- inertial force
- force that has no concrete origin
- kinetic friction
- force that opposes the motion of two systems that are in contact and moving relative to each other
- noninertial frame of reference
- accelerated frame of reference
- static friction
- force that opposes the motility of two systems that are in contact and are not moving relative to each other
- terminal velocity
- constant velocity achieved past a falling object, which occurs when the weight of the object is balanced past the upward drag forcefulness
Key Equations
| Magnitude of static friction | f s ≤ μ southward N fs≤μsN |
| Magnitude of kinetic friction | f k = μ chiliad N fk=μkN |
| Centripetal force | F c = m 5 2 r or F c = m r ω 2 Fc=mv2rorFc=mrω2 |
| Platonic angle of a banked curve | tan θ = five ii r thou tanθ=v2rg |
| Drag forcefulness | F D = 1 2 C ρ A v 2 FD=12CρAv2 |
| Stokes' police | F s = vi π r η v Fs=6πrηv |
Summary
6.1 Solving Problems with Newton's Laws
- Newton'south laws of move can exist applied in numerous situations to solve motion bug.
- Some bug contain multiple force vectors acting in different directions on an object. Exist sure to depict diagrams, resolve all force vectors into horizontal and vertical components, and depict a free-body diagram. Always analyze the management in which an object accelerates so that y'all can determine whether F net = 1000 a Fnet=ma or F net = 0 . Fnet=0.
- The normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating vertically, the normal force is less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force is always less than the full weight of the object.
- Some bug comprise several concrete quantities, such as forces, acceleration, velocity, or position. Y'all tin apply concepts from kinematics and dynamics to solve these problems.
vi.2 Friction
- Friction is a contact force that opposes the move or attempted move between two systems. Simple friction is proportional to the normal forceN supporting the two systems.
- The magnitude of static friction force between two materials stationary relative to each other is determined using the coefficient of static friction, which depends on both materials.
- The kinetic friction force between two materials moving relative to each other is adamant using the coefficient of kinetic friction, which also depends on both materials and is e'er less than the coefficient of static friction.
6.3 Centripetal Strength
- Centripetal forcefulness F ⃗ c F→c is a "middle-seeking" force that always points toward the center of rotation. Information technology is perpendicular to linear velocity and has the magnitude
- Rotating and accelerated frames of reference are noninertial. Inertial forces, such as the Coriolis forcefulness, are needed to explain motion in such frames.
6.4 Drag Force and Last Speed
- Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity in air, the drag force is determined using the elevate coefficient (typical values are given in Tabular array half-dozen.two), the expanse of the object facing the fluid, and the fluid density.
- For modest objects (such as a bacterium) moving in a denser medium (such every bit h2o), the drag force is given past Stokes' police force.
Conceptual Questions
6.ane Solving Problems with Newton's Laws
1.
To simulate the apparent weightlessness of space orbit, astronauts are trained in the hold of a cargo aircraft that is accelerating down atg. Why do they appear to be weightless, every bit measured past continuing on a bath scale, in this accelerated frame of reference? Is in that location any departure betwixt their credible weightlessness in orbit and in the shipping?
6.2 Friction
ii .
The glue on a piece of tape can exert forces. Can these forces be a type of uncomplicated friction? Explain, considering particularly that tape can stick to vertical walls and even to ceilings.
three.
When you lot learn to drive, you observe that you demand to let up slightly on the restriction pedal as you come to a finish or the car volition stop with a wiggle. Explain this in terms of the relationship between static and kinetic friction.
4 .
When you push a piece of chalk across a chalkboard, information technology sometimes screeches because it rapidly alternates between slipping and sticking to the lath. Describe this process in more particular, in particular, explaining how information technology is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.)
5.
A physics major is cooking breakfast when she notices that the frictional force betwixt her steel spatula and Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction betwixt the two materials, she quickly calculates the normal strength. What is information technology?
half-dozen.three Centripetal Force
6 .
If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you lot utilize large- or small-diameter tires? Explain.
7.
Define centripetal forcefulness. Can any blazon of force (for instance, tension, gravitational strength, friction, so on) be a centripetal force? Tin can whatsoever combination of forces be a centripetal strength?
eight .
If centripetal force is directed toward the center, why do you experience that y'all are 'thrown' away from the heart as a car goes around a curve? Explicate.
9.
Race machine drivers routinely cutting corners, as shown below (Path 2). Explain how this allows the bend to be taken at the greatest speed.
10 .
Many amusement parks have rides that make vertical loops like the i shown beneath. For safety, the cars are attached to the rails in such a manner that they cannot fall off. If the auto goes over the top at only the right speed, gravity alone will supply the centripetal force. What other strength acts and what is its direction if:
(a) The car goes over the acme at faster than this speed?
(b) The car goes over the superlative at slower than this speed?
11.
What causes h2o to be removed from clothes in a spin-dryer?
12 .
As a skater forms a circle, what strength is responsible for making his plow? Employ a gratuitous-body diagram in your reply.
13.
Suppose a kid is riding on a merry-go-round at a distance near halfway between its middle and edge. She has a lunch box resting on wax newspaper, and then that there is very little friction betwixt it and the merry-become-round. Which path shown below will the lunch box take when she lets go? The tiffin box leaves a trail in the dust on the merry-go-round. Is that trail direct, curved to the left, or curved to the right? Explain your answer.
14 .
Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your auto's speed? What is the direction of the forcefulness exerted on you past the car seat?
15.
Suppose a mass is moving in a circular path on a frictionless tabular array every bit shown beneath. In Globe's frame of reference, at that place is no centrifugal force pulling the mass abroad from the center of rotation, yet in that location is a force stretching the string attaching the mass to the nail. Using concepts related to centripetal strength and Newton'south third law, explain what strength stretches the string, identifying its physical origin.
sixteen .
When a toilet is flushed or a sink is tuckered, the h2o (and other material) begins to rotate almost the bleed on the manner down. Assuming no initial rotation and a flow initially directly straight toward the bleed, explain what causes the rotation and which direction it has in the Northern Hemisphere. (Note that this is a small effect and in most toilets the rotation is caused by directional water jets.) Would the direction of rotation reverse if water were forced up the drain?
17.
A automobile rounds a curve and encounters a patch of ice with a very low coefficient of kinetic fiction. The car slides off the road. Describe the path of the car as it leaves the road.
18 .
In ane entertainment park ride, riders enter a big vertical butt and stand against the wall on its horizontal floor. The barrel is spun upward and the floor drops abroad. Riders feel every bit if they are pinned to the wall by a force something like the gravitational force. This is an inertial strength sensed and used past the riders to explain events in the rotating frame of reference of the butt. Explain in an inertial frame of reference (Earth is nearly one) what pins the riders to the wall, and identify all forces acting on them.
nineteen.
Two friends are having a conversation. Anna says a satellite in orbit is in costless autumn because the satellite keeps falling toward Globe. Tom says a satellite in orbit is not in complimentary fall considering the acceleration due to gravity is non ix.eighty m/s two ix.80m/s2 . Who do you agree with and why?
xx .
A nonrotating frame of reference placed at the center of the Sun is very most an inertial ane. Why is information technology non exactly an inertial frame?
half-dozen.4 Drag Force and Terminal Speed
21.
Athletes such as swimmers and bicyclists vesture body suits in competition. Formulate a listing of pros and cons of such suits.
22 .
Two expressions were used for the drag force experienced by a moving object in a liquid. One depended upon the speed, while the other was proportional to the foursquare of the speed. In which types of move would each of these expressions be more applicable than the other ane?
23.
Equally cars travel, oil and gasoline leaks onto the route surface. If a light rain falls, what does this do to the command of the car? Does a heavy rain make any difference?
24 .
Why tin a squirrel jump from a tree branch to the basis and run abroad undamaged, while a human being could pause a bone in such a fall?
Problems
6.1 Solving Problems with Newton'due south Laws
25.
A 30.0-kg girl in a swing is pushed to one side and held at rest by a horizontal force F ⃗ F→ and so that the swing ropes are 30.0 ° 30.0° with respect to the vertical. (a) Calculate the tension in each of the two ropes supporting the swing under these conditions. (b) Calculate the magnitude of F ⃗ . F→.
26 .
Find the tension in each of the iii cables supporting the traffic light if it weighs two.00 × x2 N.
27.
Three forces deed on an object, considered to be a particle, which moves with abiding velocity 5 = ( 3 i ˆ − 2 j ˆ ) m/s . v=(3i^−2j^)m/s. Two of the forces are F ⃗ 1 = ( 3 i ˆ + 5 j ˆ − 6 k ˆ ) Northward F→1=(3i^+5j^−6k^)N and F ⃗ 2 = ( 4 i ˆ − 7 j ˆ + 2 grand ˆ ) Northward . F→2=(4i^−7j^+2k^)N. Find the third strength.
28 .
A flea jumps past exerting a forcefulness of i.twenty × 10 −5 N 1.20×10−5N straight down on the ground. A breeze blowing on the flea parallel to the footing exerts a force of 0.500 × 10 −six N 0.500×ten−6N on the flea while the flea is even so in contact with the footing. Discover the direction and magnitude of the dispatch of the flea if its mass is 6.00 × 10 −7 kg 6.00×10−7kg . Do not neglect the gravitational forcefulness.
29.
Two muscles in the dorsum of the leg pull upwardly on the Achilles tendon, as shown below. (These muscles are called the medial and lateral heads of the gastrocnemius muscle.) Find the magnitude and management of the total force on the Achilles tendon. What type of movement could be caused by this force?
30 .
After a mishap, a 76.0-kg circus performer clings to a trapeze, which is being pulled to the side by some other circus artist, as shown hither. Calculate the tension in the two ropes if the person is momentarily motionless. Include a free-body diagram in your solution.
31.
A 35.0-kg dolphin decelerates from 12.0 to seven.fifty m/due south in 2.xxx s to join some other dolphin in play. What average force was exerted to slow the first dolphin if it was moving horizontally? (The gravitational force is balanced by the buoyant force of the water.)
32 .
When starting a human foot race, a lxx.0-kg sprinter exerts an boilerplate strength of 650 N backward on the ground for 0.800 s. (a) What is his concluding speed? (b) How far does he travel?
33.
A big rocket has a mass of ii.00 × ten 6 kg 2.00×106kg at takeoff, and its engines produce a thrust of 3.l × 10 7 Due north . three.fifty×107N. (a) Observe its initial acceleration if it takes off vertically. (b) How long does it take to reach a velocity of 120 km/h directly upward, assuming constant mass and thrust?
34 .
A basketball game player jumps directly upwardly for a brawl. To do this, he lowers his trunk 0.300 m and then accelerates through this altitude by forcefully straightening his legs. This player leaves the floor with a vertical velocity sufficient to acquit him 0.900 m in a higher place the floor. (a) Summate his velocity when he leaves the flooring. (b) Calculate his dispatch while he is straightening his legs. He goes from zero to the velocity plant in (a) in a distance of 0.300 m. (c) Calculate the force he exerts on the floor to exercise this, given that his mass is 110.0 kg.
35.
A 2.50-kg fireworks trounce is fired direct up from a mortar and reaches a top of 110.0 m. (a) Neglecting air resistance (a poor assumption, merely we volition make it for this instance), summate the trounce's velocity when it leaves the mortar. (b) The mortar itself is a tube 0.450 m long. Summate the average acceleration of the shell in the tube equally it goes from zero to the velocity found in (a). (c) What is the average force on the shell in the mortar? Express your respond in newtons and as a ratio to the weight of the shell.
36 .
A 0.500-kg spud is fired at an angle of 80.0 ° 80.0° above the horizontal from a PVC pipe used equally a "potato gun" and reaches a height of 110.0 m. (a) Neglecting air resistance, calculate the potato's velocity when it leaves the gun. (b) The gun itself is a tube 0.450 1000 long. Calculate the average acceleration of the tater in the tube as information technology goes from goose egg to the velocity found in (a). (c) What is the average force on the murphy in the gun? Limited your answer in newtons and as a ratio to the weight of the potato.
37.
An elevator filled with passengers has a mass of 1.70 × ten iii kg i.lxx×103kg . (a) The elevator accelerates upwards from rest at a charge per unit of 1.20 k/s 2 1.20m/s2 for ane.fifty s. Calculate the tension in the cable supporting the elevator. (b) The lift continues upward at constant velocity for 8.50 south. What is the tension in the cablevision during this time? (c) The elevator decelerates at a rate of 0.600 m/s 2 0.600m/s2 for 3.00 due south. What is the tension in the cable during deceleration? (d) How high has the elevator moved above its original starting point, and what is its final velocity?
38 .
A 20.0-g ball hangs from the roof of a freight car by a string. When the freight motorcar begins to move, the string makes an angle of 35.0 ° 35.0° with the vertical. (a) What is the acceleration of the freight car? (b) What is the tension in the string?
39.
A student's backpack, full of textbooks, is hung from a leap scale fastened to the ceiling of an elevator. When the elevator is accelerating downwardly at iii.8 m/s 2 3.8m/s2 , the scale reads threescore N. (a) What is the mass of the backpack? (b) What does the scale read if the lift moves upward while slowing downwards at a rate 3.8 yard/southward 2 3.8m/s2 ? (c) What does the calibration read if the lift moves upwards at constant velocity? (d) If the lift had no brakes and the cable supporting it were to break loose and so that the lift could autumn freely, what would the spring scale read?
xl .
A service elevator takes a load of garbage, mass 10.0 kg, from a floor of a skyscraper under construction, down to basis level, accelerating downward at a rate of ane.ii m/s 2 i.2m/s2 . Observe the magnitude of the force the garbage exerts on the floor of the service elevator?
41.
A roller coaster car starts from balance at the top of a track 30.0 g long and inclined at 20.0 ° 20.0° to the horizontal. Assume that friction tin can exist ignored. (a) What is the acceleration of the car? (b) How much fourth dimension elapses before it reaches the lesser of the track?
42 .
The device shown below is the Atwood'southward machine considered in Example 6.5. Assuming that the masses of the string and the frictionless pulley are negligible, (a) find an equation for the dispatch of the two blocks; (b) find an equation for the tension in the string; and (c) detect both the acceleration and tension when block ane has mass ii.00 kg and block two has mass 4.00 kg.
43.
Two blocks are connected by a massless rope as shown below. The mass of the cake on the tabular array is 4.0 kg and the hanging mass is 1.0 kg. The tabular array and the pulley are frictionless. (a) Observe the dispatch of the system. (b) Find the tension in the rope. (c) Discover the speed with which the hanging mass hits the floor if it starts from residuum and is initially located 1.0 k from the floor.
44 .
Shown below are two carts connected by a cord that passes over a small frictionless pulley. Each cart rolls freely with negligible friction. Calculate the acceleration of the carts and the tension in the cord.
45.
A two.00 kg block (mass ane) and a 4.00 kg block (mass two) are connected past a low-cal string as shown; the inclination of the ramp is forty.0 ° 40.0° . Friction is negligible. What is (a) the acceleration of each block and (b) the tension in the string?
6.2 Friction
46 .
(a) When rebuilding his auto's engine, a physics major must exert 3.00 × ten 2 3.00×102 Northward of forcefulness to insert a dry steel piston into a steel cylinder. What is the normal force between the piston and cylinder? (b) What force would he take to exert if the steel parts were oiled?
47.
(a) What is the maximum frictional force in the knee articulation of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous do, it is possible to exert forces to the joints that are easily x times greater than the weight being supported. What is the maximum force of friction nether such weather condition? The frictional forces in joints are relatively pocket-size in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces tin cause further damage and pain.
48 .
Suppose you have a 120-kg wooden crate resting on a wood floor, with coefficient of static friction 0.500 between these wood surfaces. (a) What maximum force can y'all exert horizontally on the crate without moving it? (b) If you continue to exert this force in one case the crate starts to slip, what volition its acceleration then be? The coefficient of sliding friction is known to be 0.300 for this situation.
49.
(a) If half of the weight of a small-scale one.00 × x 3 -kg 1.00×103-kg utility truck is supported by its two drive wheels, what is the maximum dispatch it tin reach on dry physical? (b) Will a metallic cabinet lying on the wooden bed of the truck slip if information technology accelerates at this rate? (c) Solve both problems assuming the truck has 4-wheel bulldoze.
50 .
A team of 8 dogs pulls a sled with waxed wood runners on wet snowfall (mush!). The dogs have average masses of 19.0 kg, and the loaded sled with its passenger has a mass of 210 kg. (a) Calculate the acceleration of the dogs starting from balance if each dog exerts an boilerplate forcefulness of 185 N backward on the snow. (b) Calculate the force in the coupling between the dogs and the sled.
51.
Consider the 65.0-kg ice skater beingness pushed by two others shown below. (a) Notice the management and magnitude of F tot , Ftot, the total forcefulness exerted on her by the others, given that the magnitudes F 1 F1 and F ii F2 are 26.four N and 18.6 North, respectively. (b) What is her initial acceleration if she is initially stationary and wearing steel-bladed skates that point in the direction of F tot ? Ftot? (c) What is her acceleration assuming she is already moving in the management of F tot ? Ftot? (Think that friction always acts in the direction contrary that of motion or attempted motility between surfaces in contact.)
52 .
Evidence that the acceleration of any object down a frictionless incline that makes an angle θ θ with the horizontal is a = g sin θ a=gsinθ . (Notation that this dispatch is contained of mass.)
53.
Show that the acceleration of any object downwardly an incline where friction behaves simply (that is, where f k = μ one thousand Northward ) fk=μkN) is a = k ( sin θ − μ one thousand cos θ ) . a=thou(sinθ−μkcosθ). Note that the acceleration is contained of mass and reduces to the expression found in the previous problem when friction becomes negligibly small ( μ k = 0 ) . (μk=0).
54 .
Summate the deceleration of a snow boarder going upward a 5.00 ° 5.00° slope, assuming the coefficient of friction for waxed wood on wet snow. The upshot of the preceding trouble may be useful, but exist careful to consider the fact that the snowfall boarder is going uphill.
55.
A motorcar at a postal service part sends packages out a chute and down a ramp to exist loaded into commitment vehicles. (a) Summate the acceleration of a box heading down a x.0 ° 10.0° slope, assuming the coefficient of friction for a parcel on waxed wood is 0.100. (b) Detect the angle of the gradient down which this box could move at a constant velocity. Yous can neglect air resistance in both parts.
56 .
If an object is to rest on an incline without slipping, so friction must equal the component of the weight of the object parallel to the incline. This requires greater and greater friction for steeper slopes. Bear witness that the maximum angle of an incline above the horizontal for which an object volition not slide down is θ = tan −1 μ south . θ=tan−1μs. Yous may use the result of the previous problem. Presume that a = 0 a=0 and that static friction has reached its maximum value.
57.
Calculate the maximum acceleration of a car that is heading down a 6.00 ° six.00° gradient (one that makes an bending of half-dozen.00 ° 6.00° with the horizontal) under the following route atmospheric condition. You may presume that the weight of the car is evenly distributed on all iv tires and that the coefficient of static friction is involved—that is, the tires are not allowed to sideslip during the deceleration. (Ignore rolling.) Calculate for a motorcar: (a) On dry concrete. (b) On moisture concrete. (c) On ice, assuming that μ southward = 0.100 μs=0.100 , the same as for shoes on ice.
58 .
Calculate the maximum acceleration of a car that is heading up a 4.00 ° 4.00° gradient (1 that makes an angle of 4.00 ° 4.00° with the horizontal) under the following route weather. Assume that only half the weight of the car is supported by the ii drive wheels and that the coefficient of static friction is involved—that is, the tires are not allowed to skid during the acceleration. (Ignore rolling.) (a) On dry concrete. (b) On wet physical. (c) On ice, assuming that μ s = 0.100 μs=0.100 , the same equally for shoes on ice.
59.
Repeat the preceding trouble for a car with four-wheel drive.
lx .
A freight railroad train consists of ii 8.00 × 10 5 -kg 8.00×105-kg engines and 45 cars with average masses of v.50 × 10 5 kg . 5.l×105kg. (a) What strength must each engine exert backward on the track to advance the train at a rate of v.00 × 10 −2 m / due south two 5.00×10−2m/s2 if the force of friction is 7.50 × 10 5 Northward 7.50×105N , bold the engines exert identical forces? This is non a large frictional force for such a massive arrangement. Rolling friction for trains is small, and consequently, trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the strength each exerts on the other), assuming all cars take the same mass and that friction is evenly distributed among all of the cars and engines?
61.
Consider the 52.0-kg mountain climber shown below. (a) Find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock confront to remain stationary. Assume that the forcefulness is exerted parallel to her legs. Too, assume negligible force exerted past her arms. (b) What is the minimum coefficient of friction between her shoes and the cliff?
62 .
A contestant in a winter sporting event pushes a 45.0-kg cake of ice across a frozen lake as shown below. (a) Calculate the minimum strengthF he must exert to go the cake moving. (b) What is its dispatch one time it starts to move, if that force is maintained?
63.
The contestant now pulls the cake of water ice with a rope over his shoulder at the same bending above the horizontal every bit shown beneath. Calculate the minimum forceF he must exert to get the block moving. (b) What is its acceleration once it starts to movement, if that strength is maintained?
64 .
At a mail service function, a parcel that is a twenty.0-kg box slides down a ramp inclined at 30.0 ° 30.0° with the horizontal. The coefficient of kinetic friction betwixt the box and airplane is 0.0300. (a) Discover the acceleration of the box. (b) Find the velocity of the box equally it reaches the finish of the plane, if the length of the plane is two m and the box starts at balance.
6.3 Centripetal Forcefulness
65.
(a) A 22.0-kg kid is riding a playground merry-get-round that is rotating at 40.0 rev/min. What centripetal force is exerted if he is 1.25 m from its middle? (b) What centripetal force is exerted if the merry-go-circular rotates at 3.00 rev/min and he is 8.00 m from its middle? (c) Compare each force with his weight.
66 .
Calculate the centripetal force on the end of a 100-m (radius) wind turbine blade that is rotating at 0.v rev/s. Presume the mass is iv kg.
67.
What is the ideal banking angle for a gentle turn of ane.xx-km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit?
68 .
What is the ideal speed to accept a 100.0-m-radius curve banked at a twenty.0 ° twenty.0° angle?
69.
(a) What is the radius of a bobsled turn banked at 75.0 ° 75.0° and taken at 30.0 m/due south, bold it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this dispatch seem large to you?
70 .
Function of riding a bicycle involves leaning at the correct bending when making a plough, as seen below. To be stable, the force exerted by the basis must exist on a line going through the center of gravity. The strength on the bicycle bike can exist resolved into two perpendicular components—friction parallel to the route (this must supply the centripetal strength) and the vertical normal force (which must equal the arrangement's weight). (a) Bear witness that θ θ (every bit divers as shown) is related to the speedv and radius of curvaturer of the turn in the same way as for an ideally banked roadway—that is, θ = tan −1 ( v 2 / r chiliad ) . θ=tan−1(v2/rg). (b) Calculate θ θ for a 12.0-m/s turn of radius xxx.0 g (as in a race).
71.
If a car takes a banked curve at less than the platonic speed, friction is needed to continue it from sliding toward the inside of the bend (a problem on icy mountain roads). (a) Calculate the ideal speed to take a 100.0 g radius curve banked at xv.0 ° 15.0° . (b) What is the minimum coefficient of friction needed for a frightened driver to accept the same curve at twenty.0 km/h?
72 .
Mod roller coasters have vertical loops like the one shown hither. The radius of curvature is smaller at the top than on the sides and then that the downwardly centripetal dispatch at the top volition be greater than the dispatch due to gravity, keeping the passengers pressed firmly into their seats. (a) What is the speed of the roller coaster at the top of the loop if the radius of curvature in that location is 15.0 1000 and the downward acceleration of the motorcar is 1.50thou? (b) How high above the tiptop of the loop must the roller coaster start from rest, assuming negligible friction? (c) If it actually starts v.00 m college than your answer to (b), how much energy did it lose to friction? Its mass is 1.50 × 10 3 kg 1.50×103kg .
73.
A child of mass 40.0 kg is in a roller coaster car that travels in a loop of radius vii.00 m. At bespeak A the speed of the auto is 10.0 one thousand/s, and at point B, the speed is 10.five grand/s. Presume the child is not property on and does non wear a seat belt. (a) What is the force of the automobile seat on the child at point A? (b) What is the force of the motorcar seat on the child at betoken B? (c) What minimum speed is required to proceed the child in his seat at point A?
74 .
In the simple Bohr model of the basis state of the hydrogen atom, the electron travels in a round orbit effectually a stock-still proton. The radius of the orbit is 5.28 × 10 −11 m, v.28×10−11m, and the speed of the electron is 2.xviii × 10 half-dozen 1000 / due south . 2.18×106m/south. The mass of an electron is nine.xi × x −31 kg 9.11×10−31kg . What is the force on the electron?
75.
Railroad tracks follow a round curve of radius 500.0 m and are banked at an angle of 5.0 ° 5.0° . For trains of what speed are these tracks designed?
76 .
The CERN particle accelerator is circular with a circumference of 7.0 km. (a) What is the dispatch of the protons ( m = ane.67 × 10 −27 kg ) (m=1.67×10−27kg) that move around the accelerator at five % v% of the speed of calorie-free? (The speed of calorie-free is v = iii.00 × x viii chiliad/due south . v=three.00×108m/s. ) (b) What is the force on the protons?
77.
A automobile rounds an unbanked curve of radius 65 one thousand. If the coefficient of static friction between the road and motorcar is 0.seventy, what is the maximum speed at which the auto traverse the curve without slipping?
78 .
A banked highway is designed for traffic moving at ninety.0 km/h. The radius of the curve is 310 m. What is the angle of banking of the highway?
6.iv Drag Force and Terminal Speed
79.
The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Notice the terminal velocity (in meters per second and kilometers per 60 minutes) of an lxxx.0-kg skydiver falling in a pike (headfirst) position with a expanse of 0.140 m 2 0.140m2 .
eighty .
A 60.0-kg and a 90.0-kg skydiver jump from an airplane at an altitude of vi.00 × 10 three m half-dozen.00×103m , both falling in the motorway position. Brand some assumption on their frontal areas and calculate their final velocities. How long will information technology take for each skydiver to reach the ground (assuming the time to reach concluding velocity is small)? Assume all values are accurate to three significant digits.
81.
A 560-g squirrel with a surface area of 930 cm 2 930cm2 falls from a 5.0-k tree to the basis. Estimate its terminal velocity. (Utilise a drag coefficient for a horizontal skydiver.) What volition be the velocity of a 56-kg person hit the ground, assuming no drag contribution in such a curt distance?
82 .
To maintain a constant speed, the force provided by a car'southward engine must equal the drag force plus the forcefulness of friction of the road (the rolling resistance). (a) What are the drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Elevate area is 0.70 m 2 0.70m2 ) (b) What is the elevate force at 70 km/h and 100 km/h for a Hummer H2? (Drag area is 2.44 thou 2 ) ii.44m2) Assume all values are authentic to three significant digits.
83.
By what factor does the elevate force on a car increment every bit it goes from 65 to 110 km/h?
84 .
Calculate the velocity a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the driblet to exist 4 mm, the density to be ane.00 × 10 iii kg/m 3 1.00×103kg/m3 , and the surface area to be π r ii πr2 .
85.
Using Stokes' constabulary, verify that the units for viscosity are kilograms per meter per second.
86 .
Find the terminal velocity of a spherical bacterium (diameter two.00 μm ii.00μm ) falling in water. You volition first demand to annotation that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be ane.10 × 10 3 kg/grand 3 1.10×103kg/m3 .
87.
Stokes' law describes sedimentation of particles in liquids and tin can be used to measure viscosity. Particles in liquids accomplish terminal velocity quickly. One tin mensurate the time information technology takes for a particle to autumn a certain distance and and so use Stokes' law to calculate the viscosity of the liquid. Suppose a steel brawl bearing (density seven.eight × x 3 kg/1000 3 7.eight×103kg/m3 , bore 3.0 mm) is dropped in a container of motor oil. Information technology takes 12 s to fall a altitude of 0.60 yard. Calculate the viscosity of the oil.
88 .
Suppose that the resistive force of the air on a skydiver can be approximated by f = − b 5 ii . f=−bv2. If the terminal velocity of a l.0-kg skydiver is 60.0 grand/s, what is the value ofb?
89.
A modest diamond of mass 10.0 one thousand drops from a swimmer's earring and falls through the water, reaching a terminal velocity of 2.0 m/s. (a) Bold the frictional strength on the diamond obeys f = − b v , f=−bv, what isb? (b) How far does the diamond fall before it reaches 90 percent of its terminal speed?
90 .
(a) What is the final velocity of a car originally traveling at 50.0 km/h that decelerates at a rate of 0.400 thousand/southward two 0.400m/s2 for 50.0 s? Assume a coefficient of friction of ane.0. (b) What is unreasonable virtually the result? (c) Which premise is unreasonable, or which premises are inconsistent?
91.
A 75.0-kg woman stands on a bathroom calibration in an elevator that accelerates from rest to 30.0 yard/southward in 2.00 south. (a) Summate the calibration reading in newtons and compare it with her weight. (The scale exerts an upward force on her equal to its reading.) (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?
92 .
(a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 k/s. (b) What is unreasonable about the result? (c) Which bounds are unreasonable or inconsistent?
93.
As shown beneath, if M = 5.l kg, G=five.50kg, what is the tension in string 1?
94 .
As shown beneath, if F = threescore.0 N F=60.0N and 1000 = 4.00 kg, Chiliad=4.00kg, what is the magnitude of the acceleration of the suspended object? All surfaces are frictionless.
95.
Equally shown below, if Grand = six.0 kg, Thou=6.0kg, what is the tension in the connecting string? The pulley and all surfaces are frictionless.
96 .
A small space probe is released from a spaceship. The space probe has mass 20.0 kg and contains 90.0 kg of fuel. It starts from rest in deep infinite, from the origin of a coordinate organization based on the spaceship, and burns fuel at the rate of 3.00 kg/s. The engine provides a constant thrust of 120.0 N. (a) Write an expression for the mass of the space probe as a function of time, between 0 and 30 seconds, bold that the engine ignites fuel beginning at t = 0 . t=0. (b) What is the velocity after 15.0 s? (c) What is the position of the space probe after 15.0 southward, with initial position at the origin? (d) Write an expression for the position as a part of time, for t > 30.0 s . t>30.0s.
97.
A one-half-full recycling bin has mass three.0 kg and is pushed up a xl.0 ° 40.0° incline with constant speed nether the action of a 26-N strength acting up and parallel to the incline. The incline has friction. What magnitude force must human activity up and parallel to the incline for the bin to movement downwardly the incline at constant velocity?
98 .
A child has mass half-dozen.0 kg and slides down a 35 ° 35° incline with constant speed under the action of a 34-N force acting up and parallel to the incline. What is the coefficient of kinetic friction between the child and the surface of the incline?
Additional Problems
99.
The two barges shown hither are coupled past a cable of negligible mass. The mass of the front barge is 2.00 × 10 3 kg ii.00×103kg and the mass of the rear barge is 3.00 × 10 3 kg . 3.00×103kg. A tugboat pulls the front barge with a horizontal force of magnitude 20.0 × 10 3 N, 20.0×103N, and the frictional forces of the water on the front and rear barges are 8.00 × 10 3 N eight.00×103N and 10.0 × 10 three N, 10.0×103N, respectively. Find the horizontal acceleration of the barges and the tension in the connecting cable.
100 .
If the guild of the barges of the preceding practise is reversed and so that the tugboat pulls the 3.00 × 10 3 -kg 3.00×103-kg barge with a strength of 20.0 × ten 3 N , 20.0×103N, what are the acceleration of the barges and the tension in the coupling cable?
101.
An object with mass1000 moves along theten-centrality. Its position at any time is given past ten ( t ) = p t three + q t ii x(t)=pt3+qt2 wherep andq are constants. Find the cyberspace force on this object for whatever timet.
102 .
A helicopter with mass 2.35 × 10 four kg 2.35×104kg has a position given by r ⃗ ( t ) = ( 0.020 t 3 ) i ˆ + ( 2.2 t ) j ˆ − ( 0.060 t 2 ) 1000 ˆ . r→(t)=(0.020t3)i^+(ii.2t)j^−(0.060t2)k^. Find the net force on the helicopter at t = 3.0 s . t=3.0s.
103.
Located at the origin, an electric automobile of massyard is at rest and in equilibrium. A time dependent force of F ⃗ ( t ) F→(t) is applied at time t = 0 t=0 , and its components are F x ( t ) = p + n t Fx(t)=p+nt and F y ( t ) = q t Fy(t)=qt wherep,q, andn are constants. Find the position r ⃗ ( t ) r→(t) and velocity v ⃗ ( t ) five→(t) as functions of timet.
104 .
A particle of massm is located at the origin. Information technology is at remainder and in equilibrium. A time-dependent strength of F ⃗ ( t ) F→(t) is applied at time t = 0 t=0 , and its components are F x ( t ) = p t Fx(t)=pt and F y ( t ) = due north + q t Fy(t)=n+qt wherep,q, andnorth are constants. Find the position r ⃗ ( t ) r→(t) and velocity 5 ⃗ ( t ) five→(t) every bit functions of fourth dimensiont.
105.
A 2.0-kg object has a velocity of 4.0 i ˆ m/due south 4.0i^m/due south at t = 0 . t=0. A constant resultant forcefulness of ( two.0 i ˆ + iv.0 j ˆ ) N (2.0i^+4.0j^)North then acts on the object for 3.0 due south. What is the magnitude of the object'due south velocity at the finish of the iii.0-due south interval?
106 .
A 1.5-kg mass has an acceleration of ( 4.0 i ˆ − 3.0 j ˆ ) k/southward 2 . (4.0i^−three.0j^)m/s2. Simply two forces deed on the mass. If ane of the forces is ( 2.0 i ˆ − 1.4 j ˆ ) North, (2.0i^−1.4j^)Northward, what is the magnitude of the other force?
107.
A box is dropped onto a conveyor belt moving at 3.4 m/s. If the coefficient of friction betwixt the box and the chugalug is 0.27, how long will it take earlier the box moves without slipping?
108 .
Shown below is a x.0-kg block being pushed by a horizontal forcefulness F ⃗ F→ of magnitude 200.0 Due north. The coefficient of kinetic friction between the ii surfaces is 0.50. Find the dispatch of the block.
109.
Equally shown below, the mass of block ane is m one = 4.0 kg, m1=4.0kg, while the mass of cake 2 is m 2 = 8.0 kg . m2=8.0kg. The coefficient of friction between m one m1 and the inclined surface is μ thousand = 0.40 . μk=0.40. What is the acceleration of the organization?
110 .
A student is attempting to motion a thirty-kg mini-fridge into her dorm room. During a moment of inattention, the mini-fridge slides down a 35 degree incline at constant speed when she applies a strength of 25 Northward acting upwards and parallel to the incline. What is the coefficient of kinetic friction betwixt the fridge and the surface of the incline?
111.
A crate of mass 100.0 kg rests on a rough surface inclined at an angle of 37.0 ° 37.0° with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its nowadays country, the crate is just ready to slip and start to move down the airplane. The coefficient of friction is 80 % eighty% of that for the static example. (a) What is the coefficient of static friction? (b) What is the maximum force that can be applied upwards along the plane on the rope and non move the block? (c) With a slightly greater applied force, the block volition slide up the plane. Once it begins to motion, what is its acceleration and what reduced strength is necessary to continue it moving upward at constant speed? (d) If the block is given a slight nudge to get it started down the plane, what volition be its acceleration in that direction? (eastward) In one case the cake begins to slide downward, what upward force on the rope is required to continue the block from accelerating downwards?
112 .
A car is moving at high speed along a highway when the driver makes an emergency braking. The wheels become locked (stop rolling), and the resulting skid marks are 32.0 meters long. If the coefficient of kinetic friction between tires and road is 0.550, and the acceleration was constant during braking, how fast was the car going when the wheels became locked?
113.
A crate having mass 50.0 kg falls horizontally off the back of the flatbed truck, which is traveling at 100 km/h. Find the value of the coefficient of kinetic friction betwixt the route and crate if the crate slides 50 k on the road in coming to balance. The initial speed of the crate is the same equally the truck, 100 km/h.
114 .
A 15-kg sled is pulled across a horizontal, snow-covered surface by a force applied to a rope at xxx degrees with the horizontal. The coefficient of kinetic friction between the sled and the snowfall is 0.twenty. (a) If the strength is 33 N, what is the horizontal acceleration of the sled? (b) What must the forcefulness be in order to pull the sled at abiding velocity?
115.
A 30.0-g ball at the end of a cord is swung in a vertical circle with a radius of 25.0 cm. The rotational velocity is 200.0 cm/southward. Observe the tension in the string: (a) at the height of the circle, (b) at the lesser of the circle, and (c) at a distance of 12.five cm from the center of the circle ( r = 12.v cm ) . (r=12.5cm).
116 .
A particle of mass 0.fifty kg starts moves through a circular path in thexy-plane with a position given past r ⃗ ( t ) = ( 4.0 cos 3 t ) i ˆ + ( 4.0 sin 3 t ) j ˆ r→(t)=(four.0cos3t)i^+(iv.0sin3t)j^ wherer is in meters andt is in seconds. (a) Find the velocity and acceleration vectors as functions of time. (b) Evidence that the acceleration vector always points toward the centre of the circle (and thus represents centripetal acceleration). (c) Find the centripetal force vector as a function of fourth dimension.
117.
A stunt cyclist rides on the interior of a cylinder 12 m in radius. The coefficient of static friction betwixt the tires and the wall is 0.68. Find the value of the minimum speed for the cyclist to perform the stunt.
118 .
When a body of mass 0.25 kg is attached to a vertical massless spring, it is extended five.0 cm from its unstretched length of four.0 cm. The body and spring are placed on a horizontal frictionless surface and rotated about the held end of the spring at 2.0 rev/s. How far is the jump stretched?
119.
Railroad tracks follow a circular curve of radius 500.0 m and are banked at an bending of 5.00 ° five.00° . For trains of what speed are these tracks designed?
120 .
A plumb bob hangs from the roof of a railroad car. The auto rounds a round track of radius 300.0 m at a speed of 90.0 km/h. At what angle relative to the vertical does the plumb bob hang?
121.
An airplane flies at 120.0 one thousand/s and banks at a 30 ° 30° bending. If its mass is 2.l × ten 3 kg, 2.50×103kg, (a) what is the magnitude of the lift strength? (b) what is the radius of the turn?
122 .
The position of a particle is given by r ⃗ ( t ) = A ( cos ω t i ˆ + sin ω t j ˆ ) , r→(t)=A(cosωti^+sinωtj^), where ω ω is a constant. (a) Show that the particle moves in a circumvolve of radiusA. (b) Calculate d r ⃗ / d t dr→/dt and and so prove that the speed of the particle is a constant A ω . Aω. (c) Determine d 2 r ⃗ / d t 2 d2r→/dt2 and show thata is given by a c = r ω 2 . ac=rω2. (d) Summate the centripetal force on the particle. [Hint: For (b) and (c), you lot volition need to utilise ( d / d t ) ( cos ω t ) = − ω sin ω t (d/dt)(cosωt)=−ωsinωt and ( d / d t ) ( sin ω t ) = ω cos ω t . (d/dt)(sinωt)=ωcosωt.
123.
2 blocks connected past a string are pulled across a horizontal surface by a forcefulness applied to i of the blocks, every bit shown below. The coefficient of kinetic friction between the blocks and the surface is 0.25. If each block has an acceleration of 2.0 g/s 2 two.0m/s2 to the right, what is the magnitudeF of the applied force?
124 .
As shown below, the coefficient of kinetic friction betwixt the surface and the larger block is 0.20, and the coefficient of kinetic friction between the surface and the smaller block is 0.30. If F = x N F=10N and One thousand = 1.0 kg K=1.0kg , what is the tension in the connecting string?
125.
In the figure, the coefficient of kinetic friction betwixt the surface and the blocks is μ thou . μk. If Yard = 1.0 kg, Thou=one.0kg, detect an expression for the magnitude of the acceleration of either cake (in terms ofF, μ k , μk, andk).
126 .
Ii blocks are stacked equally shown beneath, and rest on a frictionless surface. There is friction between the two blocks (coefficient of friction μ μ ). An external force is practical to the top block at an angle θ θ with the horizontal. What is the maximum forceF that tin can exist applied for the two blocks to move together?
127.
A box rests on the (horizontal) dorsum of a truck. The coefficient of static friction between the box and the surface on which information technology rests is 0.24. What maximum distance can the truck travel (starting from rest and moving horizontally with constant acceleration) in three.0 south without having the box slide?
128 .
A double-incline plane is shown below. The coefficient of friction on the left surface is 0.30, and on the right surface 0.sixteen. Calculate the acceleration of the organization.
Challenge Bug
129.
In a later chapter, you volition detect that the weight of a particle varies with altitude such that due west = m grand r 0 ii r 2 westward=mgr02r2 where r 0 r0 is the radius of Globe andr is the altitude from World'due south centre. If the particle is fired vertically with velocity v 0 v0 from Earth's surface, determine its velocity equally a function of positionr. (Hint: apply a d r = v d five , adr=vdv, the rearrangement mentioned in the text.)
130 .
A big centrifuge, like the one shown beneath, is used to betrayal aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries. (a) At what angular velocity is the centripetal acceleration 10one thousand if the rider is fifteen.0 yard from the center of rotation? (b) The passenger's cage hangs on a pin at the stop of the arm, allowing information technology to swing outward during rotation as shown in the lesser accompanying figure. At what angle θ θ beneath the horizontal volition the cage hang when the centripetal acceleration is 10chiliad? (Hint: The arm supplies centripetal force and supports the weight of the cage. Depict a free-body diagram of the forces to see what the bending θ θ should be.)
131.
A car of mass 1000.0 kg is traveling along a level road at 100.0 km/h when its brakes are applied. Calculate the stopping distance if the coefficient of kinetic friction of the tires is 0.500. Neglect air resistance. (Hint: since the altitude traveled is of involvement rather than the time,x is the desired contained variable and nont. Use the Chain Rule to change the variable: d 5 d t = d five d x d 10 d t = v d five d x . ) dvdt=dvdxdxdt=vdvdx.)
132 .
An airplane flying at 200.0 m/s makes a turn that takes 4.0 min. What banking concern angle is required? What is the percentage increment in the perceived weight of the passengers?
133.
A skydiver is at an distance of 1520 m. Afterwards 10.0 seconds of free autumn, he opens his parachute and finds that the air resistance, F D FD , is given by the formula F D = − b v , FD=−bv, whereb is a constant andv is the velocity. If b = 0.750 , b=0.750, and the mass of the skydiver is 82.0 kg, first gear up differential equations for the velocity and the position, and and so find: (a) the speed of the skydiver when the parachute opens, (b) the distance fallen before the parachute opens, (c) the terminal velocity after the parachute opens (find the limiting velocity), and (d) the fourth dimension the skydiver is in the air after the parachute opens.
134 .
In a tv set commercial, a modest, spherical bead of mass 4.00 g is released from rest at t = 0 t=0 in a canteen of liquid shampoo. The terminal speed is observed to be ii.00 cm/s. Find (a) the value of the constantb in the equation v = m 1000 b ( 1 − e − b t / m ) , five=mgb(1−due east−bt/yard), and (b) the value of the resistive strength when the bead reaches terminal speed.
135.
A boater and motor gunkhole are at balance on a lake. Together, they take mass 200.0 kg. If the thrust of the motor is a constant force of 40.0 N in the direction of motion, and if the resistive force of the water is numerically equivalent to 2 times the speed5of the gunkhole, set and solve the differential equation to find: (a) the velocity of the boat at fourth dimensiont; (b) the limiting velocity (the velocity afterwards a long time has passed).
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