Mechanics

The fastest served tennis ball, served by “Big Bill” Tilden in 1931, was measured at 73.14 m/s. The mass of a tennis ball is 57 g, and the ball, which starts from rest, is typically in contact with the tennis racquet for 30.0 ms. Assuming constant acceleration, (a) what force did Big Bill’s tennis racquet exert on the ball if he hit it essentially horizontally? (b) Draw free-body diagrams of the ball during the serve and just after it moved free of the racquet.

Two crates, one with mass 4.00 kg and the other with mass 6.00 kg, sit on the frictionless surface of a frozen pond, connected by a light rope (Fig. P4.39). A woman wearing golf shoes (for traction) pulls horizontally on the 6.00-kg crate with a force F that gives the crate an acceleration of $\mathbf{2}\mathbf{.}\mathbf{50}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. (a) What is the acceleration of the 4.00 kg crate? (b) Draw a free-body diagram for the 4.00 kg crate. Use that diagram and Newton’s second law to find the tension  in the rope that connects the two crates. (c) Draw a free-body diagram for the 6.00 kg crate. What is the direction of the net force on the 6.00-kg crate? Which is larger in magnitude,T or F ? (d) Use part (c) and Newton’s second law to calculate the magnitude of F.

Two blocks connected by a light horizontal rope sit at rest on a horizontal, frictionless surface. Block A has mass , and block B has mass . A constant horizontal force F=60.0 N is applied to block A (Fig. P4.40). In the first 5.00 s after the force is applied, block A moves 18.0 m to the right. (a) While the blocks are moving, what is the tension  in the rope that connects the two blocks? (b) What is the mass of block B?

A 6.50 kg instrument is hanging by a vertical wire inside a spaceship that is blasting off from rest at the earth’s surface. This spaceship reaches an altitude of 276 m in 15.0 s with constant acceleration. (a) Draw a free-body diagram for the instrument during this time. Indicate which force is greater. (b) Find the force that the wire exerts on the instrument.

The froghopper (Philaenus spumarius), the champion leaper of the insect world, has a mass of  and leaves the ground (in the most energetic jumps) at 4.0 m/s from a vertical start. The jump itself lasts a mere 1.0 ms before the insect is clear of the ground. Assuming constant acceleration, (a) draw a free-body diagram of this mighty leaper during the jump; (b) find the force that the ground exerts on the froghopper during the jump; and (c) express the force in part (b) in terms of the froghopper’s weight.

A loaded elevator with very worn cables has a total mass of 2200 kg, and the cables can withstand a maximum tension of 28000 N. (a) Draw the free-body force diagram for the elevator. In terms of the forces on your diagram, what is the net force on the elevator? Apply Newton’s second law to the elevator and find the maximum upward acceleration for the elevator if the cables are not to break. (b) What would be the answer to part (a) if the elevator were on the moon, where $\mathbf{g}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{62}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$?

After an annual checkup, you leave your physician’s office, where you weighed 683 N . You then get into an elevator that, conveniently, has a scale. Find the magnitude and direction of the elevator’s acceleration if the scale reads (a) 725 N and (b) 595 N.

A nail in a pine board stops a 4.9-N  hammer head from an initial downward velocity of 3.2 m/s in a distance of 0.45 cm. In addition, the person using the hammer exerts a 15-N downward force on it. Assume that the acceleration of the hammer head is constant while it is in contact with the nail and moving downward. (a) Draw a free-body diagram for the hammer head. Identify the reaction force for each action force in the diagram. (b) Calculate the downward force $\overrightarrow{\mathbf{F}}$ exerted by the hammer head on the nail while the hammer head is in contact with the nail and moving downward. (c) Suppose that the nail is in hardwood and the distance the hammer head travels in coming to rest is only 0.12 cm. The downward forces on the hammer head are the same as in part (b). What then is the force $\overrightarrow{\mathbf{F}}$ exerted by the hammer head on the nail while the hammer head is in contact with the nail and moving downward?

A 75.0-kg man steps off a platform 3.10 m above the ground. He keeps his legs straight as he falls, but his knees begin to bend at the moment his feet touch the ground; treated as a particle, he moves an additional  0.60 m before coming to rest. (a) What is his speed at the instant his feet touch the ground? (b) If we treat the man as a particle, what is his acceleration (magnitude and direction) as he slows down, if the acceleration is assumed to be constant? (c) Draw his free body diagram. In terms of the forces on the diagram, what is the net force on him? Use Newton’s laws and the results of part (b) to calculate the average force his feet exert on the ground while he slows down. Express this force both in newtons and as a multiple of his weight.

The two blocks in Fig. P4.48 are connected by a heavy uniform rope with a mass of 4.00 kg . An upward force of 200 N is applied as shown. (a) Draw three free-body diagrams: one for the 6.00 kg block, one for the 4.00 kg rope, and another one for the 5.00 kg block. For each force, indicate what body exerts that force. (b) What is the acceleration of the system? (c) What is the tension at the top of the heavy rope? (d) What is the tension at the midpoint of the rope?

Boxes A and B are connected to each end of a light vertical rope (Fig. P4.49). A constant upward force $\mathbf{F}\mathbf{=}\mathbf{80}\mathbf{.}\mathbf{0}\mathbf{N}$ is applied to box A. Starting from rest, box B descends $\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$ in $\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{s}$. The tension in the rope connecting the two boxes is $\mathbf{36}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{N}$. What are the masses of (a) box B, (b) box A?

You have landed on an unknown planet, Newtonian, and want to know what objects weigh there. When you push a certain tool, starting from rest, on a frictionless horizontal surface with a $\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{-}\mathbf{N}$ force, the tool moves $\mathbf{16}\mathbf{.0 }\mathbf{m}$ in the first $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{s}$. You next observe that if you release this tool from rest at $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$ above the ground, it takes $\mathbf{2}\mathbf{.}\mathbf{58}\mathbf{ }\mathbf{s}$ to reach the ground. What does the tool weigh on Newtonian, and what does it weigh on Earth?

 A mysterious rocket-propelled object of mass $\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{kg}$ is initially at rest in the middle of the horizontal, frictionless surface of an ice-covered lake. Then a force-directed east and with magnitude $\mathbf{F}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\left(16.8N/s\right)$ is applied. How far does the object travel in the first $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{s}$ after the force is applied?

 The position of a training helicopter $\left(\mathrm{weight} 2.75\times {10}^{5}N\right)$ in a test is given by $\hat{\mathbf{r}}\mathbf{=}\left(0.020 m/s^3\right){\mathbf{t}}^{\mathbf{3}}\hat{\mathbf{i}}\mathbf{+}\left(2.2 m/s\right)\mathbf{t}\hat{\mathbf{j}}\mathbf{-}\left(0.060 m/s^2\right){\mathbf{t}}^{\mathbf{2}}\hat{\mathbf{k}}$. Find the net force on the helicopter at $\mathbf{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$.

The table* gives automobile performance data for a few types of cars:

(a) During an acceleration of 0 to $\mathbf{60} \mathbf{mph}$, which car has the largest average net force acting on it? The smallest? (b) During this acceleration, for which car would the average net force on a  $\mathbf{72}.\mathbf{0}-\mathbf{kg}$passenger be the largest? The smallest? (c) When the Ferrari F430 accelerates from  0to 100 mph in$\mathbf{8}.\mathbf{6} \mathbf{s}$, what is the average net force acting on it? How does this net force compare with the average net force during the acceleration from 0 to $\mathbf{60} \mathbf{mph}$? Explain why these average net forces might differ. (d) Discuss why a car has a top speed. What is the net force on the Ferrari F430 when it is traveling at its top speed, $\mathbf{196} \mathbf{mph}$?

An 8.00 kg box sits on a level floor. You give the box a sharp push and find that it travels 8.22 m in 2.8 s
 before coming to rest again. (a) You measure that with a different push the box traveled 4.20 m in 2.0 s. Do you think the box has a constant acceleration as it slows down? Explain your reasoning. (b) You add books to the box to increase its mass. Repeating the experiment, you give the box a push and measure how long it takes the box to come to rest and how far the box travels. The results, including the initial experiment with no added mass, are given in the table:

In each case, did your push give the box the same initial speed? What is the ratio between the greatest initial speed and the smallest initial speed for these four cases? (c) Is the average horizontal force f exerted on the box by the floor the same in each case? Graph the magnitude of force f versus the total mass m of the box plus its contents, and use your graph to determine an equation for f as a function of m.

Forces on a Dancer’s Body. Dancers experience large forces associated with the jumps they make. For example, when a dancer lands after a vertical jump, the force exerted on the head by the neck must exceed the head’s weight by enough to cause the head to slow down and come to rest. The head is about $\mathbf{9}.\mathbf{4}\%$ of a typical person’s mass. Video analysis of a $\mathbf{65}-\mathbf{kg}$ dancer landing after a vertical jump shows that her head decelerates from $\mathbf{4}.\mathbf{0}\mathbf{m}/\mathbf{s}$ to rest in a time of  $\mathbf{0}.\mathbf{20}\mathbf{s}$.

The forces on a dancer can be measured directly when a dancer performs a jump on a force plate that measures the force between her feet and the ground. A graph of force versus time throughout a vertical jump performed on a force plate is shown in Fig. P4.60. What is happening at $\mathbf{0}.\mathbf{4}\mathbf{s}$ ? The dancer is 

(a) bending her legs so that her body is accelerating downward; 

(b) pushing her body up with her legs and is almost ready to leave the ground; (c) in the air and at the top of her jump; 

(d) landing and her feet have just touched the ground.

You are a Starfleet captain going boldly where no man has gone before. You land on a distant planet and visit an engineering testing lab. In one experiment a short, light rope is attached to the top of a block and a constant upward force $\mathbf{ }\mathbf{F}$ is applied to the free end of the rope. The block has mass m and is initially at rest. As $\mathbf{F}$ is varied, the time for the block to move upward 
$\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$ is measured. The values that you collected are given in the table:

(a) Plot $\mathbf{F}$ versus the acceleration   of the block. (b) Use your graph to determine the mass   of the block and the acceleration of gravity $\mathbf{g}$ at the surface of the planet. Note that even on that planet, measured values contain some experimental error.

An object of mass $\mathbf{m}$ is at rest in equilibrium at the origin. At $\mathbf{t}\mathbf{=}\mathbf{0}$ a new force $\overrightarrow{\mathbf{F}}\left(t\right)$  is applied that has components ${\mathbf{F}}_{\mathbf{x}}\left(t\right)\mathbf{=}{\mathbf{k}}_{\mathbf{1}}\mathbf{+}{\mathbf{k}}_{\mathbf{2}}\mathbf{y}$ and ${\mathbf{F}}_{\mathbf{y}}\left(t\right)\mathbf{=}{\mathbf{k}}_{\mathbf{3}}\mathbf{t}$ . Where, ${\mathbf{k}}_{\mathbf{1}}$ , ${\mathbf{k}}_{\mathbf{2}}$ , and ${\mathbf{k}}_{\mathbf{3}}$ are constants. Calculate the position $\overrightarrow{\mathbf{r}}\left(t\right)$ and velocity  $\overrightarrow{\mathbf{v}}\left(t\right)$ vectors as functions of time.

Forces on a Dancer’s Body. Dancers experience large forces associated with the jumps they make. For example, when a dancer lands after a vertical jump, the force exerted on the head by the neck must exceed the head’s weight by enough to cause the head to slow down and come to rest. The head is about 9.4% of a typical person’s mass. Video analysis of a$\mathbf{65}\mathbf{-}\mathbf{kg}$ dancer landing after a vertical jump shows that her head decelerates from 4.0m/s to rest in a time of 0.20s.

While the dancer is in the air and holding a fixed pose, what is the magnitude of the average force that her neck exerts on her head? (a) 0N ; (b) 60N ; (c) 120N ; (d) 180N.

Dancers experience large forces associated with the jumps they make. For example, when a dancer lands after a vertical jump, the force exerted on the head by the neck must exceed the head’s weight by enough to cause the head to slow down and come to rest. The head is about 9.4% of a typical person’s mass. Video analysis of a$\mathbf{65}\mathbf{-}\mathbf{kg}$ dancer landing after a vertical jump shows that her head decelerates from 4.0m/s to rest in a time of 0.20s.

Compared with the force her neck exerts on her head during the landing, the force her head exerts on her neck is (a) the same; (b) greater; (c) smaller; (d) greater during the first half of the landing and smaller during the second half of the landing.

A man sits in a seat that is hanging from a rope. The rope passes over a pulley suspended from the ceiling, and the man holds the other end of the rope in his hands. What is the tension in the rope, and what force does the seat exert on him? Draw a free-body force diagram for the man.

“In general, the normal force is not equal to the weight.” Give an example in which these two forces are equal in magnitude, and at least two examples in which they are not.

If there is a net force on a particle in uniform circular motion, why doesn’t the particle’s speed change?

The centrifugal force is not included in the free-body diagrams of Figs. 5.34b and 5.35. Explain why not.

To push a box up a ramp, which requires less force: pushing horizontally or pushing parallel to the ramp? Why?

A woman in an elevator lets go of her briefcase, but it does not fall to the floor. How is the elevator moving?

A block rests on an inclined plane with enough friction to prevent it from sliding down. To start the block moving is it easier to push it up the plane or down the plane? Why?

A crate slides up an inclined ramp and then slides down the ramp after momentarily stopping near the top. There is kinetic friction between the surface of the ramp and the crate. Which is greater? (i) The crate’s acceleration going up the ramp; (ii) the crate’s acceleration going down the ramp; (iii) both are the same. Explain.

A crate of books rests on a level floor. To move it along the floor at a constant velocity, why do you exert less force if you pull it at an angle above the horizontal than if you push it at the same angle below the horizontal?

When you stand with bare feet in a wet bathtub, the grip feels fairly secure, and yet a catastrophic slip is quite possible. Explain this in terms of the two coefficients of friction.

You are pushing a large crate from the back of a freight elevator to the front as the elevator is moving to the next floor. In which situation is the force you must apply to move the crate the least, and in which is it the greatest: when the elevator is accelerating upward, when it is accelerating downward, or when it is traveling at constant speed? Explain

It is often said that “friction always opposes motion.” Give at least one example in which (a) static friction causes motion, and (b) kinetic friction causes motion.

A curve in a road has a bank angle calculated and posted for  $\mathbf{80}\mathbf{km}\mathbf{/}\mathbf{h}$. However, the road is covered with ice, so you cautiously plan to drive slower than this limit. What might happen to your car? Why?

You swing a ball on the end of a lightweight string in a horizontal circle at constant speed. Can the string ever be truly horizontal? If not, would it slope above the horizontal or below the horizontal? Why?

A professor swings a rubber stopper in a horizontal circle on the end of a string in front of his class. He tells Caroline, in the front row, that he is going to let the string go when the stopper is directly in front of her face. Should Caroline worry?

To keep the forces on the riders within allowable limits, many loop-the-loop roller coaster rides are designed so that the loop is not a perfect circle but instead has a larger radius of curvature at the bottom than at the top. Explain.

A tennis ball drops from rest at the top of a tall glass cylinder—first with the air pumped out of the cylinder so that there is no air resistance, and again after the air has been readmitted to the cylinder. You examine multiflash photographs of the two drops. Can you tell which photo belongs to which drop? If so, how?

You throw a baseball straight upward with speed ${\mathbf{v}}_{\mathbf{0}}$. When the ball returns to the point from where you threw it, how does its speed compare to ${\mathbf{v}}_{\mathbf{0}}$ (a) in the absence of air resistance and (b) in the presence of air resistance? Explain.

You throw a baseball straight upward. If you do not ignore air resistance, how does the time required for the ball to reach its maximum height compare to the time required for it to fall from its maximum height back down to the height from which you threw it? Explain.

You have two identical tennis balls and fill one with water. You release both balls simultaneously from the top of a tall building. If air resistance is negligible, which ball will strike the ground first? Explain. What if air resistance is not negligible?

A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in Fig. Q5.25 best represents its acceleration as a function of time?

A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in Fig. Q5.26 best represents its vertical velocity component as a function of time?

When a batted baseball moves with air drag when does the ball travel a greater horizontal distance? (i) While climbing to its maximum height; (ii) while descending from its maximum height back to the ground; (iii) the same for both? Explain in terms of the forces acting on the ball.

In Fig. E5.2 each of the suspended blocks has weight w. The pulleys are frictionless, and the ropes have negligible weight. In each case, draw a free-body diagram and calculate the tension T in the rope in terms of w.

 A 75.0-kg wrecking ball hangs from a uniform, heavy-duty chain of mass 26.0 kg. 

(a) Find the maximum and minimum tensions in the chain. 

(b) What is the tension at a point three-fourths of the way up from the bottom of the chain?

In the treatment of spine injuries, it is often necessary to provide tension along the spinal column to stretch the backbone. One device for doing this is the Stryker frame (Fig. E5.4a). A weight W is attached to the patient (sometimes around a neck collar, Fig. E5.4b), and friction between the person’s body and the bed prevents sliding.

(a) If the coefficient of static friction between a 78.5-kg patient’s body and the bed is 0.75, what is the maximum traction force along the spinal column that W can provide without causing the patient to slide? 

(b) Under the conditions of maximum traction, what is the tension in each cable attached to the neck collar?

A picture frame hung against a wall is suspended by two wires attached to its upper corners. If the two wires make the same angle with the vertical, what must this angle be if the tension in each wire is equal to 0.75 of the weight of the frame? (Ignore any friction between the wall and the picture frame.).

A large wrecking ball is held in place by two light steel cables (Fig. E5.6). If the mass m of the wrecking ball is 3620 kg, what are 

(a) the tension T_B in the cable that makes an angle of 40° with the vertical and 

(b) the tension T_A in the horizontal cable?

Find the tension in each cord in Fig. E5.7 if the weight of the suspended object is w.