Mechanics

Can the total work done on an object during a displacement be negative? Explain. If the total work is negative, can its magnitude be larger than the initial kinetic energy of the object? Explain.

A net force acts on an object and accelerates it from rest to speed V₁. In doing so, the force does an amount of work W₁. By what factor must the work done on the object be increased to produce three times the final speed, with the object again starting from rest? 

A truck speeding down the highway has a lot of kinetic energy relative to a stopped state trooper but no kinetic energy relative to the truck driver. In these two frames of reference, is the same amount of work required to stop the truck? Explain. 

You are holding a briefcase by the handle, with your arm straight down by your side. Does the force your hands exert do work on the briefcase when (a) you walk at a constant speed down a horizontal halfway and (b) you ride an escalator from the first to second floor of a building? In both cases justify your answer. 

When a book slides over a tabletop, the force of friction does negative work on it. Can friction ever do positive work? Explain. (Hint: Think of a box in the back of an accelerating truck.) 

Time yourself while running up a flight of steps, and compute the average rate at which you do work against the force of gravity. Explain your answer in watts and in horsepower.

Fractured Physics. Many terms from Physics are badly misused in everyday language. In both cases, explain the errors involved. (a) A strong person is called powerful. What is wrong with this use of power? (b) When a worker carries a bag of concrete along a level construction site, people say he did a lot of work. Did He?

An advertisement for a portable electrical generating unit claims that the unit’s diesel engine produces 28,000 hp to drive an electrical generator that produces 30 MW of electrical power. Is this possible?. Explain.

A car speeds up while the engine delivers constant power. Is the acceleration greater at the beginning of this process or at the end? Explain.

Consider a graph of instantaneous power versus time, with the vertical P-axis starting at P = 0. What is the physical significance of the area under the P-versus-t curve between vertical lines at ${\mathbf{t}}_{\mathbf{1}}$ and ${\mathbf{t}}_{\mathbf{2}}$? How could you find the average power from the graph? Draw a P-versus-t curve that consists of two straight line sections and for which the peak power is equal to twice the average power.

A nonzero net force acts on an object. Is it possible for any of the following quantities to be constant: the object’s (a) speed; (b) velocity (c) Kinetic energy.

When a certain force is applied to an ideal spring, the spring stretches a distance x from its unstretched length and does work W. If instead twice the work is applied, what distance (in terms of x) does the spring stretch from its unstretched length, and how much work (in terms of W) is required to stretch it this distance?

When a certain force is applied to an ideal spring, the spring stretches a distance x from its unstretched length and does work W. If instead twice the work is applied, what distance (in terms of x) does the spring stretch from its unstretched length, and how much work (in terms of W) is required to stretch it this distance?

if work W is required to stretch a spring distance x from its unstretched length, what work ( in terms of W) is required to stretch the spring an additional distance x?

A loaded grocery cart is rolling across a parking lot in a strong wind. You apply a constant force $\overrightarrow{\mathbf{F}}\mathbf{=}\left(30N\right)\hat{\mathbf{i}}\mathbf{‐}\left(40N\right)\hat{\mathbf{j}}$ to the cart as it undergoes a displacement  $\overrightarrow{\mathbf{s}}\mathbf{=}\left(‐9.0m\right)\hat{\mathbf{i}}\mathbf{‐}\left(3.0m\right)\hat{\mathbf{j}}$. How much work does the force you apply do on the grocery cart?

A 8.00-kg ball is tied to the end of a string 1.60 m long and swung in a vertical circle. (a) During one complete circle, starting anywhere, calculate the total work done on the ball by (i) the tension in the string and (ii) gravity. (b) Repeat part (a) for motion along the semicircle from the lowest to the highest point on the path.

A 12.0-kg package in a mail-sorting room slides 2.00 m down a chute that is inclined at 53.0°

  below the horizontal. The coefficient of kinetic friction between the package and the chute’s surface is 0.40. Calculate the work done on the package by (a) friction, (b) gravity, and (c) the normal force, (d) What is the net work done on the package.

A 128.0 N carton is pulled up a frictionless baggage ramp inclined at 30.0⁰ above the horizontal by a rope exerting a 72.0 N pull parallel to the ramp’s surface. If the carton travels 5.20 m along the surface of the ramp, calculate the work done on it by (a) the rope, (b) by gravity, and (c) the normal force of the ramp. (d) What is the net work done on the carton?

Question: A 128.0 N carton is pulled up a frictionless baggage ramp inclined at 30.0^o above the horizontal by a rope exerting a 72.0 N pull parallel to the ramp’s surface. If the carton travels 5.20 m along the surface of the ramp, calculate the work done on it by (a) the rope, (b) by gravity, and (c) the normal force of the ramp. (d) What is the net work done on the carton?

A boxed 10.0-kg computer monitor is dragged by friction 5.50 m upward along a conveyor belt inclined at an angle of  $\mathbf{36}\mathbf{.}\mathbf{9}\mathbf{°}$ above the horizontal. If the monitor’s speed is a constant 2.10 cm/s, how much work is done on the monitor (a) by friction, (b) by gravity, and (c) the normal force of the conveyor belt.

 A large crate sits on the floor of a warehouse. Paul and Bob also apply constant horizontal forces to the crate. The force applied by Paul has magnitude 48.0 N and direction $\mathbf{61}\mathbf{.}\mathbf{0}\mathbf{°}$ south of west. How much work does Paul’s force do during a displacement of the crate that is 12.0 m in the direction  $\mathbf{22}\mathbf{°}$ east of north?

You apply a constant force $\mathbf{F}\mathbf{=}\left(‐68.0N\right)\mathbf{ }\hat{\mathbf{i}}\mathbf{+}\left(36.0N\right)\hat{\mathbf{j}}$  to the 380-kg car as the car travels 48.0 m in a direction that is 240⁰ counter clockwise from the +x-axis. How much work does the force you apply do on the car? 

On a farm, you are pushing on a stubborn pig with a constant horizontal force with a magnitude 30.0 N and a direction 37.0⁰ counterclockwise from the +x-axis. How much work does this force do during a displacement of the pig that is (a) $\mathbf{s}\mathbf{=}\left(5.00\right)\hat{\mathbf{i}}$ ; (b)  $\mathbf{s}\mathbf{=}\left(‐6.00\right)\hat{\mathbf{j}}$; (c) $\mathbf{s}\mathbf{=}\mathbf{‐}\left(2.00\right)\hat{\mathbf{i}}\mathbf{+}\left(5.00\right)\hat{\mathbf{j}}$ .

A 1.50-kg book is sliding along a rough horizontal surface. At point A it is moving at 3.21 m/s, and at point B it has slowed to 1.25 m/s. (a) How much work was done on the book between A and B? (b) If -0.750 J of work is done on the book from B to C, how fast it is moving at point C? (c) How fast it would be moving at C +0.750 J of work was done on it from B to C?

Bio Animal Energy. Adult cheetahs, the fastest of the great cats, have a mass of about 70 kg and have been clocked to run at 72 mi/h (32m/s). (a) How many joules of kinetic energy does such a swift cheetah have? (b) By what factor would its kinetic energy change if its speed were doubled?

Some typical kinetic energies. (a) In the Bohr model of the atom, the ground-state electron in hydrogen has an orbital speed of 2190 km/s. What is its kinetic energy (consult Appendix F) (b) If you drop a 1.0-kg weight (about 2 lb) from a height of 1.0 m, how many joules of kinetic energy will it have when it reaches the ground? (c) Is it reasonable that a 30.0-kg child could run fast enough to have 100 J of kinetic energy?

Meteor crater. About 50,000 years ago, a meteor crashed into the earth near present-day Flagstaff Arizona. Measurements from 2005 estimate that this meteor had a mass of about 1.4 X 10⁸ kg (around 15,000 tons) and hit the ground at a speed of 12 km/s. (a) How much kinetic energy did this meteor deliver to the ground? (b) How does this energy compare to the energy released by a 1.0-megaton nuclear bomb? (A megaton bomb releases the same amount of energy as a million tons of TNT, and 1.0 ton of TNT releases 4.184 X 10⁹ J of energy).

A 4.80-kg watermelon is dropped from rest from the roof of an 18.0-m tall building and feels no appreciable air resistance. (a) calculate the work done by gravity on the watermelon during its displacement from the roof to the ground. (b) Just before it strikes the ground, what is the watermelon’s (i) Kinetic energy and (ii) speed? (c) Which of the answers in parts (a) and (b) would be different if there were appreciable air resistance? 

Use the work-energy theorem to solve each of these problems. You can use Newton’s laws to check your answers. Neglect air resistance in all cases. (A) A branch falls from the top of a 95.0-m- tall redwood tree, starting from rest. How fast is it moving when it reaches the ground? (b) A volcano ejects a boulder directly upward 525 m into the air. How fast was the boulder moving just as it left the volcano?

Use the work-energy theorem to solve each of these problem. You can use Newton’s laws to check your answers. (a) A skier is moving at 5.00 m/s encounters a long, rough horizontal patch of snow having a coefficient of kinetic friction of 0.220 with her skis. How far does she travel on this patch before stopping? (b) Suppose the rough patch in part (a) was only 2.90 m long. How fats would the skier be moving when she reached the end of the patch? (c) At the base of a frictionless icy hill that rises at 25.0⁰ above the horizontal, a toboggan has a speed of 12.0 m/s toward the hill. How high vertically above the base will it go before stopping?  

You are a member of an Alpine rescue team. You must project a box of supplies up an incline of constant slope angle  so that it reaches a standard skier who is a vertical distance h above the bottom of the incline. The incline is slippery, but there is some friction present with kinetic friction coefficient  ${\mathbf{\mu }}_{\mathbf{k}}$ . Use the the work energy theorem to calculate the minimum speed you must give the box at the bottom of the incline so that it will reach the skier. Express your answer in terms of g, h${\mathbf{\mu }}_{\mathbf{k}}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{\alpha }$.

You are a member of an Alpine rescue team. You must project a box of supplies up an incline of constant slope angle  so that it reaches a standard skier who is a vertical distance h above the bottom of the incline. The incline is slippery, but there is some friction present with kinetic friction coefficient  ${\mathbf{\mu }}_{\mathbf{k}}$ . Use the the work energy theorem to calculate the minimum speed you must give the box at the bottom of the incline so that it will reach the skier. Express your answer in terms of  g, h ${\mathbf{\mu }}_{\mathbf{k}}$ and $\mathbf{\alpha }$ .

You throw a 3.00-N rock vertically into the air from ground level. You observe that when it is 15.0 m above the ground , it is travelling at 25.0 m/s. Use the work-energy theorem to find (a) the rock’s speed just as it left the ground and (b) its maximum height.

A sled with mass 12 kg moves in a straight line on a frictionless, horizontal surface. At one point in its path, its speed is 4 m/s; after it has traveled 2.50 m beyond this point, its speed is 6.00 m/s. Use the work–energy theorem to find the force acting on the sled, assuming that this force is constant and that it acts in the direction of the sled’s motion.

Suppose the sled in Exercise 6.36 is initially at rest at x = 0. Use the work–energy theorem to find the speed of the sled at 

(a) x = 8.0 m and 

(b) x = 12.0 m. Ignore friction between the sled and the surface of the pond.

 A spring of force constant $\mathbf{300}\mathbf{.0}\mathbf{\text{\hspace{0.17em}N/m}}$and unstretched length 0.240 m is stretched by two forces, pulling in opposite directions at opposite ends of the spring, that increase to 15.0 N. How long will the spring now be, and how much work was required to stretch it that distance?

A $\mathbf{6}\mathbf{.0}\mathbf{\text{\hspace{0.17em}kg}}$box moving at $\mathbf{3}\mathbf{.0}\mathbf{\text{\hspace{0.17em}m/s}}$on a horizontal, frictionless surface runs into a light spring of force constant$\mathbf{75}\mathbf{.0}\mathbf{\text{\hspace{0.17em}N/cm}}$. Use the work-energy theorem to find the maximum compression of the spring.

As part of your daily workout, you lie on your back and push with your feet against a platform attached to two stiff springs arranged side by side so that they are parallel to each other. When you push the platform, you compress the springs. You do$\mathbf{80}\mathbf{.0}\mathbf{\text{\hspace{0.17em}J}}$ of work when you compress the springs$\mathbf{0}\mathbf{.200}\mathbf{\text{\hspace{0.17em}m}}$ from their uncompressed length. 

(a) What magnitude of force must you apply to hold the platform in this position? 

(b) How much additional work must you do to move the platform$\mathbf{0}\mathbf{.200}\mathbf{\text{\hspace{0.17em}m}}$ farther, and what maximum force must you apply?

(a) In Example 6.7 (Section 6.3), it was calculated that with the air track turned off, the glider travels 8.6 cm before it stops instantaneously. How large would the coefficient of static friction ${\mathit{\mu }}_{\mathbf{s}}$have to be to keep the glider from springing back to the left? 

(b) If the coefficient of static friction between the glider and the track is ${\mathit{\mu }}_{\mathbf{s}}\mathbf{=}\mathbf{0}\mathbf{.60}$, what is the maximum initial speed${\mathit{v}}_{\mathbf{1}}$ that the glider can be given and still remain at rest after it stops instantaneously? With the air track turned off, the coefficient of kinetic friction is${\mathit{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.47}$ .

A mass m slides down a smooth inclined plane from an initial vertical height h, making an angle a with the horizontal. 

(a) The work done by a force is the sum of the work done by the components of the force. Consider the components of gravity parallel and perpendicular to the surface of the plane. Calculate the work done on the mass by each of the components, and use these results to show that the work done by gravity is exactly the same as if the mass had fallen straight down through the air from a height h. 

(b) Use the work–energy theorem to prove that the speed of the mass at the bottom of the incline is the same as if the mass had been dropped from height h, independent of the angle a of the incline. Explain how this speed can be independent of the slope angle. 

(c) Use the results of part (b) to find the speed of a rock that slides down an icy frictionless hill, starting from rest 15.0 m above the bottom.

A 12-pack of Omni-Cola (mass 4.30 kg) is initially at rest on a horizontal floor. It is then pushed in a straight line for by a trained dog that exerts a horizontal force with magnitude 36.0 N. Use the work–energy theorem to find the final speed of the 12-pack if 

(a) there is no friction between the 12-pack and the floor, and 

(b) the coefficient of kinetic friction between the 12-pack and the floor is 0.30.

A soccer ball with mass 0.420 kg is initially moving with speed 2 m/s. A soccer player kicks the ball, exerting a constant force of magnitude 40.0 N in the same direction as the ball’s motion. Over what distance must the player’s foot be in contact with the ball to increase the ball’s speed to  6.00 m/s

A little red wagon with mass 7.00 kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of 4 m/s and then is pushed 3.0 m in the direction of the initial velocity by a force with a magnitude of 10.0 N. (a) Use the work energy theorem to calculate the wagon’s final speed.

(b) Calculate the acceleration produced by the force. Use this acceleration in the kinematic relationships of Chapter 2 to calculate the wagon’s final speed. Compare this result to that calculated in part (a)

A block of ice with mass 2.00 kg slides 1.35 m down an inclined plane that slopes downward at an angle of 36.9° below the horizontal. If the block of ice starts from rest, what is its final speed?

A car is traveling on a level road with speed ${\mathit{v}}_{\mathbf{0}}$ at the instant when the brakes lock, so that the tires slide rather than roll.

(a) Use the work–energy theorem to calculate the minimum stopping distance of the car in terms of ${\mathit{v}}_{\mathbf{0}}$ , g, and the coefficient of kinetic friction ${\mathit{\mu }}_{\mathbf{k}}$ between the tires and the road. 

(b) By what factor would the minimum stopping distance change if (i) the coefficient of kinetic friction were doubled, or (ii) the initial speed were doubled, or (iii) both the coefficient of kinetic friction and the initial speed were doubled?

A 30.0-kg crate is initially moving with a velocity that has magnitude 3.90 m/s in a direction 37.0° west of north. How much work must be done on the crate to change its velocity to 5.62m/s in a direction 63.0° south of east?

A surgeon is using material from a donated heart to repair a patient’s damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a 16.0-cm strip of the donated aorta reveal that it stretches 3.75 cm when a 1.50-N pull is exerted on it. 

(a) What is the force constant of this strip of aortal material? 

(b) If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is 1.14 cm, what is the greatest force it will be able to exert there?

To stretch a spring 3.00 cm from its unstretched length, 12.0 J of work must be done. 

(a) What is the force constant of this spring? 

(b) What magnitude force is needed to stretch the spring 3.00 cm from its unstretched length? 

(c) How much work must be done to compress this spring 4.00 cm from its unstretched length, and what force is needed to compress it this distance?

Three identical 8.50-kg masses are hung by three identical springs (Fig. E6.35). Each spring has a force constant of 7.80 kN/m and was 12.0 cm long before any masses were attached to it. 

(a) Draw a free-body diagram of each mass. 

(b) How long is each spring when hanging as shown? (Hint: First isolate only the bottom mass. Then treat the bottom two masses as a system. Finally, treat all three masses as a system.)

A child applies a force $\overrightarrow{\mathbf{F}}$ parallel to the x-axis to a 10.0kg sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the x-component of the force she applies varies with the x-coordinate of the sled as shown in Fig. E6.36. Calculate the work done by $\overrightarrow{\mathbf{F}}$ when the sled moves

(a) from $\mathit{x}\mathbf{=}\mathbf{0}$ to $\mathit{x}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$;

(b) from $\mathbf{x}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$ to $\mathbf{x}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$; 

(c) from $\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{ }\mathbf{m}$ to $\mathbf{x}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$.