Mechanics

A 1130-kg car is held in place by a light cable on a very smooth (frictionless) ramp (Fig. E5.8). The cable makes an angle of 31.0° above the surface of the ramp, and the ramp itself rises at 25.0° above the horizontal. 

(a) Draw a free-body diagram for the car. 

(b) Find the tension in the cable. 

(c) How hard does the surface of the ramp push on the car?

A man pushes on a piano with mass 180 kg; it slides at constant velocity down a ramp that is inclined at 19.0° above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes 

(a) parallel to the incline and 

(b) parallel to the floor.

In Fig. E5.10 the weight w is 60.0 N. 

(a) What is the tension in the diagonal string? 

(b) Find the magnitudes of the horizontal forces F₁ and F₂ that must be applied to hold the system in the position shown.

“A ball is thrown from the edge of a high cliff. Regardless of the angle at which it is thrown, due to air resistance, the ball will eventually end up moving vertically downward.” Justify this statement.”

An astronaut is inside a $\mathbf{2}\mathbf{.}\mathbf{25}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }\mathbf{kg}$  rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound (331 m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than 4g.

(a) What is the maximum initial thrust this rocket’s engines can have but just barely avoid blackout? Start with a free-body diagram of the rocket. 

(b) What force, in terms of the astronaut’s weight w, does the rocket exert on her? Start with a free-body diagram of the astronaut. 

(c) What is the shortest time it can take the rocket to reach the speed of sound?

A rocket of initial mass 125 kg (including all the contents) has an engine that produces a constant vertical force (the thrust) of 1720 N. Inside this rocket, a 15.5-N electrical power supply rests on the floor. 

(a) Find the initial acceleration of the rocket.  

(b) When the rocket initially accelerates, how hard does the floor push on the power supply? (Hint: Start with a free-body diagram for the power supply.)

On September 8, 2004, the Genesis spacecraft crashed in the Utah desert because its parachute did not open. The 210-kg capsule hit the ground at 311 km/h and penetrated the soil to a depth of 81.0 cm. 

(a) What was its acceleration (in m/s ² and in g’s), assumed to be constant, during the crash? 

(b) What force did the ground exert on the capsule during the crash? Express the force in newtons and as a multiple of the capsule’s weight. 

(c) How long did this force last?

Three sleds are being pulled horizontally on frictionless horizontal ice using horizontal ropes (Fig. E5.14). The pull is of magnitude 190N. Find (a) the acceleration of the system and (b) the tension in ropes A and B.

Atwood’s Machine. A 15.0kg load of bricks hangs from one end of a rope that passes over a small, frictionless pulley. A 28.0kg counterweight is suspended from the other end of the rope (Fig. E5.15). The system is released from rest. (a) Draw two free-body diagrams, one for the load of bricks and one for the counterweight. (b) What is the magnitude of the upward acceleration of the load of bricks? (c) What is the tension in the rope while the load is moving? How does the tension compare to the weight of the load of bricks? To the weight of the counterweight?

An 8.00-kg block of ice, released from rest at the top of a 1.50-m  long frictionless ramp, slides downhill, reaching a speed of 2.50 m/s  at the bottom. (a) What is the angle between the ramp and the horizontal? (b) What would be the speed of the ice at the bottom if the motion were opposed by a constant friction force of 10.0 N parallel to the surface of the ramp?

A light rope is attached to a block with mass 4.00 -kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass m is suspended from the other end. When the blocks are released, the tension in the rope is 15.0N . (a) Draw two free-body diagrams: one for each block. (b) What is the acceleration of either block? (c) Find m . (d) How does the tension compare to the weight of the hanging block? 

A transport plane takes off from a level landing field with two gliders in tow, one behind the other. The mass of each glider is 700 kg , and the total resistance (air drag plus friction with the runway) on each may be assumed constant and equal to 2500N . The tension in the towrope between the transport plane and the first glider is not to exceed . (a) If a speed of 40 m/s  is required for take off, what minimum length of runway is needed? (b) What is the tension in the towrope between the two gliders while they are accelerating for the take off?

A 750.0-kg boulder is raised from a quarry 125 m deep by a long uniform chain having a mass of 575 kg . This chain is of uniform strength, but at any point it can support a maximum tension no greater than 2.50 times its weight without breaking. (a) What is the maximum acceleration the boulder can have and still get out of the quarry, and (b) how long does it take to be lifted out at maximum acceleration if it started from rest?

A 550-N physics student stands on a bathroom scale in an elevator that is supported by a cable. The combined mass of student plus elevator is 850 kg. As the elevator starts moving, the scale reads 450 N. 

(a) Find the acceleration of the elevator (magnitude and direction). 

(b) What is the acceleration if the scale reads 670 N? 

(c) If the scale reads zero, should the student worry? Explain. 

(d) What is the tension in the cable in part (a) and (c)?

When jumping straight up from a crouched position, an average person can reach a maximum height of about 60 cm. During the jump, the person’s body from the knees up typically rises a distance of around $\mathbf{50}\mathbf{ }\mathbf{cm}$. To keep the calculations simple and yet get a reasonable result, assume that the entire body rises this much during the jump.(a) With what initial speed does the person leave the ground to reach a height of $\mathbf{60}\mathbf{ }\mathbf{cm}$? (b) Draw a free-body diagram of the person during the jump. (c) In terms of this jumper’s weight  , what force does the ground exert on him or her during the jump?

A 2540 kg test rocket is launched vertically from the launch pad. Its fuel (of negligible mass) provides a thrust force such that its vertical velocity as a function of time is given by $\mathit{v}\mathit{\left(}\mathit{t}\mathit{\right)}\mathit{=}\mathit{A}\mathit{t}\mathit{+}\mathit{B}\mathit{t}{\mathit{ }}^{\mathit{2}}$ , where A and B are constants and time is measured from the instant the fuel is ignited. The rocket has an upward acceleration of $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ at the instant of ignition and,1.00 s later, an upward velocity of 2.00 m/s. (a) Determine A and B, including their SI units. (b) At 4.00 s after fuel ignition, what is the acceleration of the rocket, and (c) what thrust force does the burning fuel exert on it, assuming no air resistance? Express the thrust in newton and as a multiple of the rocket’s weight. (d) What was the initial thrust due to the fuel?

A 2.00-kg box is moving to the right with speed on a horizontal, frictionless surface. At t = 0 a horizontal force is applied to the box. The force is directed to the left and has magnitude $\mathit{F}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\left(6.00 N/s^2\right){\mathit{t}}^{\mathbf{2}}$. (a) What distance does the box move from its position at t = 0 before its speed is reduced to zero? (b) If the force continues to be applied, what is the speed of the box at t = 3.00 s?

A 5.00-kg crate is suspended from the end of a short vertical rope of negligible mass. An upward force   is applied to the end of the rope, and the height of the crate above its initial position is given by $\mathit{y}\left(t\right)\mathbf{=}\left(2.80m/s\right)\mathit{t}\mathbf{+}\left(0.610m/s^3\right){\mathit{t}}^{\mathbf{3}}$ . What is the magnitude of  F when  $t=4.00s$ ?

After emergencies with major blood loss, a patient is placed in the Trendelen burg position, in which the foot of the bed is raised to get maximum blood flow to the brain. If the coefficient of static friction between atypical patient and the bed sheets is 1.20, what is the maximum angle at which the bed can be tilted with respect to the floor before the patient begins to slide?

In a laboratory experiment on friction, a 135-N block resting on a rough horizontal table is pulled by a horizontal wire. The pull gradually increases until the block begins to move and continues to increase thereafter. Figure E5.26 shows a graph of the friction force on this block as a function of the pull. (a) Identify the regions of the graph where static friction and kinetic friction occur. (b) Find the coefficients of static friction and kinetic friction between the block and the table. (c) Why does the graph slant upward at first but then level out? (d) What would the graph look like if a 135-N brick were placed on the block, and what would the coefficients of friction be?

Figure E5.26

A stockroom worker pushes a box with mass 16.8 kg on a horizontal surface with a constant speed of 3.50 m/s. The coefficient of kinetic friction between the box and the surface is 0.20. (a) What horizontal force must the worker apply to maintain the motion? (b) If the force calculated in part (a) is removed, how far does the box slide before coming to rest?

A box of bananas weighing  40.0 N rests on a horizontal surface. The coefficient of static friction between the box and the surface is 0.40 , and the coefficient of kinetic friction is 0.20 . (a) If no horizontal force is applied to the box and the box is at rest, how large is the friction force exerted on it? (b) What is the magnitude of the friction force if a monkey applies a horizontal force of  6.0 N to the box and the box is initially at rest? (c) What minimum horizontal force must the monkey apply to start the box in motion? (d) What minimum horizontal force must the monkey apply to keep the box moving at constant velocity once it has been started? (e) If the monkey applies a horizontal force of 18.0 N , what is the magnitude of the friction force and what is the box’s acceleration?

Question: A box of bananas weighing 40.0N rests on a horizontal surface. The coefficient of static friction between the box and the surface is 0.40, and the coefficient of kinetic friction is 0.20. (a) If no horizontal force is applied to the box and the box is at rest, how large is the friction force exerted on it? (b) What is the magnitude of the friction force if a monkey applies a horizontal force of 6.0N to the box and the box is initially at rest? (c) What minimum horizontal force must the monkey apply to start the box in motion? (d) What minimum horizontal force must the monkey apply to keep the box moving at constant velocity once it has been started? (e) If the monkey applies a horizontal force of 18.0N, what is the magnitude of the friction force and what is the box’s acceleration?

A 45.0-kg crate of tools rests on a horizontal floor. You exert a gradually increasing horizontal push on it, and the crate just begins to move when your force exceeds 313 N. Then you must reduce your push to 208 N to keep it moving at a steady 25.0 m/s . (a) What are the coefficients of static and kinetic friction between the crate and the floor? (b) What push must you exert to give it an acceleration of $\mathbf{1}\mathbf{.}\mathbf{10}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$? (c) Suppose you were performing the same experiment on the moon, where the acceleration due to gravity is $\mathbf{1}\mathbf{.}\mathbf{62}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. (i) What magnitude push would cause it to move? (ii) What would its acceleration be if you maintained the push in part (b)?

Question: A 45.0-kg crate of tools rests on a horizontal floor. You exert a gradually increasing horizontal push on it, and the crate just begins to move when your force exceeds 313 N. Then you must reduce your push to 208 N to keep it moving at a steady 25.0m/s. (a) What are the coefficients of static and kinetic friction between the crate and the floor? (b) What push must you exert to give it an acceleration of  $\mathbf{1}\mathbf{.}\mathbf{10}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$? (c) Suppose you were performing the same experiment on the moon, where the acceleration due to gravity is $\mathbf{1}\mathbf{.}\mathbf{62}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$. (i) What magnitude push would cause it to move? (ii) What would its acceleration be if you maintained the push in part (b)?

Some sliding rocks approach the base of a hill with a speed of 12 m/s. The hill rises at 36° above the horizontal and has coefficients of kinetic friction and static friction of 0.45 and 0.65, respectively, with these rocks. (a) Find the acceleration of the rocks as they slide up the hill. (b) Once a rock reaches its highest point, will it stay there or slide down the hill? If it stays, show why. If it slides, find its acceleration on the way down.

Question: Some sliding rocks approach the base of a hill with a speed of . The hill rises at 36° above the horizontal and has coefficients of kinetic friction and static friction of 0.45 and 0.65, respectively, with these rocks. (a) Find the acceleration of the rocks as they slide up the hill. (b) Once a rock reaches its highest point, will it stay there or slide down the hill? If it stays, show why. If it slides, find its acceleration on the way down.

A box with mass 10.0 kg moves on a ramp that is inclined at an angle $\mathbf{55}\mathbf{°}$of above the horizontal. The coefficient of kinetic friction between the box and the ramp surface is ${\mathit{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{300}$. Calculate the magnitude of the acceleration of the box if you push on the box with a constant force F = 120.0 N that is parallel to the ramp surface and (a) directed down the ramp, moving the box down the ramp; (b) directed up the ramp, moving the box up the ramp.

Question: A box with mass 10.0 kg moves on a ramp that is inclined at an angle of above the horizontal. The coefficient of kinetic friction between the box and the ramp surface is . Calculate the magnitude of the acceleration of the box if you push on the box with a constant force that is parallel to the ramp surface and (a) directed down the ramp, moving the box down the ramp; (b) directed up the ramp, moving the box up the ramp.

A pickup truck is carrying a toolbox, but the rear gate of the truck is missing. The toolbox will slide out if it is set moving. The coefficients of kinetic friction and static friction between the box and the level bed of the truck are 0.355 and 0.650 , respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to 30.0 m/s  without causing the box to slide? Draw a free-body diagram of the toolbox.

Question: A pickup truck is carrying a toolbox, but the rear gate of the truck is missing. The toolbox will slide out if it is set moving. The coefficients of kinetic friction and static friction between the box and the level bed of the truck are 0.355 and 0.650, respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to 30.0m/s without causing the box to slide? Draw a free-body diagram of the toolbox.

You are lowering two boxes, one on top of the other, down a ramp by pulling on a rope parallel to the surface of the ramp (Fig. E5.33). Both boxes move together at a constant speed of $15.0 \text{cm/s}$ . The coefficient of kinetic friction between the ramp and the lower box is $\mathbf{0}.\mathbf{444}$, and the coefficient of static friction between the two boxes is $\mathbf{0}.800$. (a) What force do you need to exert to accomplish this? (b) What are the magnitude and direction of the friction force on the upper box?

Figure E5.33

Consider the system shown in Fig. E5.34. Block A weighs 45.0 N, and block B weighs 25.0 N . Once block B is set into downward motion, it descends at a constant speed. (a) Calculate the coefficient of kinetic friction between block A and the tabletop. (b) A cat, also of weight , falls asleep on top of block A. If block B is now set into downward motion, what is its acceleration (magnitude and direction)?

Question: Consider the system shown in Fig. E5.34. Block A weighs 45.0N, and block B weighs 25.0N. Once block B is set into downward motion, it descends at a constant speed. (a) Calculate the coefficient of kinetic friction between block A and the tabletop. (b) A cat, also of weight 45.0N, falls asleep on top of block A. If block B is now set into downward motion, what is its acceleration (magnitude and direction)?

You throw a baseball straight upward. The drag force is proportional to $v^2$ . In terms of g, what is the y-component of the ball’s acceleration when the ball’s speed is half its terminal speed and 

 (a) it is moving up?

 (b) It is moving back down?

A stone with mass 0.80 kg is attached to one end of a string 0.90 m long. The string will break if its tension exceeds60.0 N. The stone is whirled in a horizontal circle on a frictionless tabletop; the other end of the string remains fixed. 

(a) Draw a free body diagram of the stone. 

(b) Find the maximum speed the stone can attain without the string breaking.

A small remote-controlled car with mass 1.60 kg moves at a constant speed of $v=12.0\text{\hspace{0.17em}m/s}$ in a track formed by a vertical circle inside a hollow metal cylinder that has a radius of 5.00 m(Fig. E5.45). What is the magnitude of the normal force exerted on the car by the walls of the cylinder at (a) point A (bottom of the track) and (b) point B (top of the track)?

A 1125-kg car and a 2250-kg pickup truck approach a curve on a highway that has a radius of 225 m. 

(a) At what angle should the highway engineer bank this curve so that vehicles traveling at $65.0\text{\hspace{0.17em}mi/h}$ can safely round it regardless of the condition of their tires? Should the heavy truck go slower than the lighter car?

(b) As the car and truck round the curve at $65.0\text{\hspace{0.17em}mi/h}$, find the normal force on each one due to the highway surface.

(a) If the coefficient of kinetic friction between tires and dry pavement is 0.80 , what is the shortest distance in which you can stop a car by locking the brakes when the car is traveling at 28.7 m/s (about 65mi/h )? (b) On wet pavement the coefficient of kinetic friction may be only 0.25 . How fast should you drive on wet pavement to be able to stop in the same distance as in part (a)? (Note: Locking the brakes is not the safest way to stop.)

A 25.0 kg box of textbooks rests on a loading ramp that makes an angle   with the horizontal. The coefficient of kinetic friction is 0.25 , and the coefficient of static friction is 0.35 . (a) As   is increased, find the minimum angle at which the box starts to slip. (b) At this angle, find the acceleration once the box has begun to move. (c) At this angle, how fast will the box be moving after it has slid 5.0 m along the loading ramp?

Question: A 25.0kg box of textbooks rests on a loading ramp that makes an angle $\mathit{\alpha }$ with the horizontal. The coefficient of kinetic friction is 0.25, and the coefficient of static friction is 0.35. (a) As $\mathit{\alpha }$ is increased, find the minimum angle at which the box starts to slip. (b) At this angle, find the acceleration once the box has begun to move. (c) At this angle, how fast will the box be moving after it has slid 5.0m along the loading ramp?

Two crates connected by a rope lie on a horizontal surface (Fig. E5.37). Crate A has mass m_A, and crate B has mass m_B. The coefficient of kinetic friction between each crate and the surface is ${\mathbf{\mu }}_{\mathbf{k}}$. The crates are pulled to the right at constant velocity by a horizontal force $\overrightarrow{\mathbf{F}}$. Draw one or more free-body diagrams to calculate the following in terms of m_A, m_B and ${\mathbf{\mu }}_{\mathbf{k}}$: (a) the magnitude of $\overrightarrow{\mathbf{F}}$ and (b) the tension in the rope connecting the blocks.

Figure E5.37

A box with mass m is dragged across a level floor with coefficient of kinetic friction  ${\mathit{\mu }}_{\mathbf{k}}$ by a rope that is pulled upward at an angle $\mathit{\theta }$ above the horizontal with a force of magnitude  F . (a) In terms of m , ${\mathit{\mu }}_{\mathbf{k}}$,$\mathit{\theta }$ and g , obtain an expression for the magnitude of the force required to move the box with constant speed. (b) Knowing that you are studying physics, a CPR instructor asks you how much force it would take to slide a 90 - kg  patient across a floor at constant speed by pulling on him at an angle of  $\mathbf{25}\mathbf{°}$ above the horizontal. By dragging weights wrapped in an old pair of pants down the hall with a spring balance, you find that ${\mathit{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{35}$ . Use the result of part (a) to answer the instructor’s question

As shown in Fig. E5.34, block A (mass 2.25 kg) rests on a tabletop. It is connected by a horizontal cord passing over a light, frictionless pulley to a hanging block B (mass 1.30kg ). The coefficient of kinetic friction between block A and the tabletop is 0.450. The blocks are released then from rest. Draw one or more free-body diagrams to find (a) the speed of each block after they move 3.00 cm and (b) the tension in the cord

A large crate with mass   rests on a horizontal floor. The coefficients of friction between the crate and the floor are ${\mathit{\mu }}_{\mathbf{s}}$ and ${\mathit{\mu }}_{\mathbf{k}}$ . A woman pushes downward with a force $\overrightarrow{\mathbf{F}}$ on the crate at an angle $\mathit{\theta }$ below the horizontal. (a) What magnitude of force  $\overrightarrow{\mathbf{F}}$ is required to keep the crate moving at constant velocity? (b) If ${\mathit{\mu }}_{\mathbf{s}}$ is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. Calculate this critical value of ${\mathit{\mu }}_{\mathbf{s}}$ .

A 52-kg ice skater spins about a vertical axis through her body with her arms horizontally out stretched; she makes 2.0 turns each second. The distance from one hand to the other is 1.50 m. Biometric measurements indicate that each hand typically makes up about 1.25% of body weight. (a) Draw a free-body diagram of one of the skater’s hands. (b) What horizontal force must her wrist exert on her hand? (c) Express the force in part (b) as a multiple of the weight of her hand.

Question: A 52-kg ice skater spins about a vertical axis through her body with her arms horizontally out stretched; she makes 2.0 turns each second. The distance from one hand to the other is 1.50 m. Biometric measurements indicate that each hand typically makes up about 1.25% of body weight. (a) Draw a free-body diagram of one of the skater’s hands. (b) What horizontal force must her wrist exert on her hand? (c) Express the force in part (b) as a multiple of the weight of her hand.

A small car with mass 0.800kg travels at constant speed on the inside of a track that is a vertical circle with radius 5.00m (Fig. E5.45). If the normal force exerted by the track on the car when it is at the top of the track (point B) is 6.00N , what is the normal force on the car when it is at the bottom of the track (point A)?

Question: A small model car with mass m  travels at constant speed on the inside of a track that is a vertical circle with radius (Fig. E5.45). If the normal force exerted by the track on the car when it is at the bottom of the track (point A) is equal to 250 Mg ,how much time does it take the car to complete one revolution around the track?

Figure E5.45

A flat (unbanked) curve on a highway has a radius of 170.0 m. A car rounds the curve at a speed of 25.0 m/s . (a) What is the minimum coefficient of static friction that will prevent sliding? (b) Suppose that the highway is icy and the coefficient of static friction between the tires and pavement is only one-third of what you found in part (a). What should be the maximum speed of the car so that it can round the curve safely?