AP Physics Assignment – Work and Energy

Objectives/HW

 The student will be able to: HW: 1 Define and apply the concept of work (and the Joule) for constant or varying force and solve related problems. 1 – 9 2 Define and apply kinetic energy.  State and apply the work-energy theorem and solve related problems. 10 – 15 3 Solve problems using conservation of mechanical energy, including situations involving nonconservative forces. 16 – 23 4 Solve problems involving gravitational potential energy in which g is not taken to be constant. 24 – 26 5 Solve problems involving work and energy for a mass attached to a spring. 27 – 29 6 Define and apply the concepts of conservative force and potential energy and solve related problems. 30 – 32 7 Define and apply the concept of power (and the Watt) and solve related problems. 33 – 37 8 Solve problems involving machines and efficiency. 38 – 40

Homework Problems

1.      A kid mows the lawn with a push mower.  He exerts a force of 50.0 N along the direction of the mower’s handle, angled 30.0° relative to horizontal.  This force moves the mower a distance of 8.00 m over level ground at constant speed.  (a) Find the work done on the mower by the kid.  (b) Find the work done on the mower by friction.  (c) Find the work done on the mower by gravity.

2.      A weightlifter is “working out” with a barbell that has a total mass of 50.0 kg.  Determine the amount of work done on the barbell for the following movements:  (a) the bar is bench pressed upward 0.450 m, (b) the bar is lowered 0.450 m, (c) the bar is dead lifted 0.900 m.

3.      A man of mass 80.0 kg rides the step of an escalator from the bottom floor to the top floor of the mall.  The man is moved 7.00 m, 35.0° relative to the bottom floor.  (a) Find the work done by the normal force.  (b) Find the work done by gravity.

4.      In order to burn off one calorie of food energy a person must do at least 4190 J of work (actually maybe ten times that amount).  (a) How many steps of height 17 cm must a person that weighs 750 N climb in order to do 4190 J work?  (b) What mass object would the person have to carry up one step in order to do 4190 J work?

5.      A physics instructor marks on a chalkboard with a 51-gram piece of chalk.  He moves the chalk from point A to point B three times – each time along a different path as shown below.  Calculate the total work done by gravity acting on the chalk for each of the following:  (a) the path with two segments, (b) the diagonal path (use W = Fdcosf), and (c) the longest path.

problems 5 – 7:

6.      Suppose the chalk starts at point B and is moved along any one of the paths to point A and then returned along any one of the other paths to point B.  Show that the total work done by gravity along such a “loop” or “circuit” is zero.

7.      Suppose the instructor pushes the chalk against the board with force 5.0 N and μ = 0.30.  Find the work done by friction:  (a) from A to B along the shortest path, (b) from A to B along the longest path, and (c) from A around the perimeter of the figure and back to A.

8.      The amount of force associated with the stretching of an elastic band can be modeled by F = ax + bx3, where x = amount of stretch, a = 5.0 N/m, and b = 12 N/m3.  (a) Find the amount of work a person has to do to stretch the band from x = 0 to x = 0.10 m.  (b) Find the work done by the elastic band as it is stretched from x = 0.10 m to x = 0.20 m.

9.      A spring with constant k = 48.0 N/m is hung from the ceiling.  A mass of 0.500 kg is attached to the free end of the spring and slowly lowered to a point of equilibrium, at which the mass will hang without moving.  (a) Find the work done on the mass by the spring.  (b) Find the work done on the mass by gravity.  (c) Find the work done on the mass by the person that lowered it.  (d) How much work must be done to pull the mass 20.0 cm farther down?

10.  A skydiver with mass 85.0 kg drops out of a hovering helicopter and falls 700.0 m before pulling the ripcord and releasing his parachute.  (a) Find the work done by gravity.  (b) What would be the speed just before opening the chute if there was no air resistance?  (c) If the actual speed just before the chute opens is 50.0 m/s, what was the average force of air resistance?  (d) Once the parachute is open, the air does work in amount −175 kJ as the speed is decreased to 6.00 m/s.  What distance does the skydiver fall while this happens?

11.  A bullet with mass 5.0 grams is moving horizontally with speed 375 m/s when it hits a tree.  The bullet penetrates the tree 10.0 cm.  Find the average force of the bullet on the wood.

12.  A block of wood with mass 0.50 kg is initially at rest on a floor where μk = 0.30.  A person pushes the block with a horizontal force of 4.0 N and lets go.  The block slides a total distance of 2.0 m from beginning to end.  Find the maximum speed of the block.

13.  Using one gallon of gasoline the engine/transmission of a certain small car can do 45.8 MJ of work driving the car forward.  Find the distance it can travel on one gallon of gas at constant speed in the following circumstances:  (a) total friction = 568 N (driver only, speed = 56 mph); (b) total friction = 662 N (driver and 3 passengers, speed = 56 mph); (c) total friction = 879 N (driver and 3 passengers, speed = 78 mph).

14.  Now suppose the same car with driver only (m = 1000 kg) accelerates from 0 to 25.0 m/s.  During this acceleration the car moves 200 m and there is 508 N average friction.  (a) Find the work done by the engine/transmission.  (b) Find the average forward force on the car.  (c) How many times can this be done using one gallon of gas and what total distance is this?

15.  Based on the results of the previous two problems discuss the factors that influence gas mileage and describe the means by which we can reduce the gas needed for transportation.

16.  A kid throws a rock with mass 155 g off a bridge and into the water below.  The rock is initially 10.0 m above the water and has a speed of 20.0 m/s.  (a) Find the speed with which the rock hits the water.  (b) Is this result dependent on the direction that the rock is thrown?  Explain.  (c) Suppose the size of the splash depends on the energy of the rock.  Determine specifically two means by which the kid could conceivable double the size of the splash (calculate the necessary changes).

17.  A bowling ball with a mass of 6.00 kg is hung from the ceiling with a rope and set into motion as a pendulum.  As it swings its height above the floor varies from 1.50 m to 0.800 m.  (a) Find the total energy of the ball relative to the floor.  (b) Find the maximum speed of the ball as it swings back and forth.  (c) If the rope breaks, what would be the speed of the ball hitting the floor?  (d) Would the impact with the floor depend on when the rope breaks?  Discuss.

18.  A small ball is tied to a string and twirled in a vertical circle of radius 30.0 cm.  Ignore friction.  When the ball reaches the lowest point in the circle it has velocity 5.00 m/s, east.  (a) Find the velocity and acceleration at the highest point in the circle.  (b) Find the velocity and acceleration when the string is horizontal and the ball is coming down.

19.  A very small block is placed on top of a large hemispherical surface as shown below.  The block is placed ever so slightly to one side of the highest part of the surface and slides without friction.  Solve for the angle θ at which the block will fly off the surface and into the air.

20.  A rollercoaster has an initial hill that leads to a circular loop of radius R.  (a) Show that the top of the hill must be at least ˝ R higher than the highest part of the loop.  (b) Discuss additional factors that must be considered in the design of an actual rollercoaster of this type.

21.  A roofer accidentally dislodges a block of wood.  The block slides 3.00 m along the roof where μ = 0.300 and then flies off the edge and falls to the ground.  The roof is tilted 30.0 ° above horizontal and the edge of the roof is 5.00 m above ground level.  (a) Find the impact speed of the block.  (b) Contemplate the steps that would be required to solve this problem without the use of energy concepts (it could be done).  Would the result be the same?  Discuss.

22.  A certain ramp is 3.00 m long and 2.00 m high.  A block that weighs 15.0 N is released from rest at the top and slides to the bottom, ending with speed 5.00 m/s.  (a) Find the work done by friction.  (b) Find the force of friction.  (c) Find the amount of work and the amount of force necessary to push the block back up the ramp.

23.  A mover needs to put a 20.0 kg box into the van.  The box is on the ground at the end of a ramp leading into the van.  The ramp is 3.00 m long and 1.20 m high.  (a) Find the work the mover does on the box if it is pushed up the ramp assuming μ = 0.300.  (b) Find the work the mover does on the box if he simply lifts it 1.20 m and carries it to the van.  (c) There are advantages and disadvantages for each method – explain.

24.  Consider the largest asteroid Ceres:  m = 9.43 × 1020 kg, r = 470 km.  (a) Find the escape speed of Ceres.  (b) If a bullet were fired upward from the surface with speed 450 m/s, how high would it go?  (c) If a bullet were fired horizontally from the surface at speed 550 m/s (and encountered no obstacles) what would happen to it?  (d) A good jumper can leave the ground with speed 4.0 m/s.  How high could such a person jump from the surface of Ceres?

25.  The Chandra X-Ray Telescope, mass 5400 kg, occupies an elliptical orbit such that its distance from Earth’s center varies from 16 030 km at perigee to 145 600 km at apogee.  Its speed at apogee is 737.2 m/s.  (a) Find the total energy of Chandra.  (b) Find its speed at perigee.  (c) Find its speed when it is at its mean distance from the center of Earth.

26.  Suppose a satellite with mass 500.0 kg is to be placed in a circular orbit at an altitude 450 km above Earth.  (a) Find the speed required to orbit at that altitude.  (b) Find the amount of work that must be done on the satellite to put it into this orbit.  Ignore air resistance.

27.  A bungee jumper of mass 90.0 kg drops from a high bridge attached to an elastic cord with original length 40.0 m.  The bungee can be modeled with a spring constant k = 175 N/m.  (a) Find how far the person falls before being stopped by the bungee cord.  (b) Find the maximum speed that occurs during the fall.  (c) Describe all of the energy transformations and/or transfers that occur during this event.

28.  A spring with constant k hangs from the ceiling.  A mass m is hooked to the free end of the spring and moved downward a certain distance and released.  (a) Solve for the maximum distance that the mass can be moved downward and released such that it will not come unhooked.  (b) Solve for the maximum speed of the mass such that it does not come unhooked.  Note:  the mass may become unhooked if it goes above the resting position of the spring’s free end.

29.  A block of mass 0.750 kg is free to move on a horizontal surface where μ = 0.250.  As shown in the diagram below, the block is placed against a spring with constant k = 83.0 N/m.  The spring is compressed 0.100 m and then the block is released.  (a) Find the speed of the block just as it leaves the spring.  (b) Find the total distance the block will slide.  (c) Describe all of the energy transformations and/or transfers that occur during this event.

30.  The potential energy of an atom in a diatomic molecule can be modeled by:  , where x = the separation of the two atoms.  (a) Determine the force acting on an atom as a function of separation.  (b) Find the separation at which the potential energy is minimized.  (c) Find the work required to separate the two atoms (i.e. the binding energy).  (d) Sketch a single graph showing two curves:  U and F as functions of x. (Hint:  Use a graphing calculator to see the shape of the curves – a and b are any convenient positive values.)

31.  An object with mass 3.00 kg moves along the x-axis and is subject to a net force in newtons given by:  F(x) = 2x − 4x3, where x is position in meters.  The object is placed at x = 1.00 m and released from rest.  (a) Determine a potential energy function U(x), using the origin as a reference.  (b) Sketch a single graph showing the two curves U and F.  (c) Describe the resulting motion of the object in words.  (d) Find the initial acceleration.  (e) Find the maximum speed.  (f) Starting at the same initial position, what initial speed would be required to propel the object to x = −2.00 m?

32.  Suppose a planet has uniform density such that gravity within the planet can be modeled by:  g(r) = GMr/R3, where M and R characterize the planet.  (a) Determine F(r) for a small mass m located somewhere in the planet.  (b) Find a potential energy function U(r) for r < R.  (c) Find the speed of an object that falls through a narrow tunnel to the center of the planet.  Ignore friction.

33.  An elevator car with mass 2750 kg is lifted 20.0 m in 11.0 s.  Find the average power output of the lift system.

34.  A Mini Cooper with mass 1300 kg and a 115 hp (86 kW) engine can go from 0 to 60 mph (26.8 m/s) in 9.8 s.  (a) Find the work done by the engine during this acceleration.  (b) Find the work done by friction.  (c) By how much would the 0 to 60 mph time be improved if the car is equipped with a 168 hp engine (the Cooper S)?

35.  A person with mass 60.0 kg runs 7.50 m up some stairs, gaining 4.00 m elevation in 3.50 s.  (a) Find the power output of the person.  (b) Find the time for the same person to carry 10.0 kg up the same stairs assuming the same power output.

36.  A certain boat with a 75 hp (1 hp = 746 W) engine has a top speed of 15 m/s through the water.  (a) Find the force of drag on the boat at its top speed.  (b) Assuming drag is directly proportional to speed, how powerful an engine on the same boat would be needed to increase the top speed to 25 m/s?

37.  The road to the top of Pike’s Peak is 31 km long and climbs 2260 meters in elevation.  A car and driver, mass 1000 kg, travels this road at an average speed of 15 m/s with friction in amount 450 N.  (a) Find the power output of the car’s drive train.  (b) If the maximum power output is 100 hp, what would be the shortest road (and therefore steepest) it could possibly climb to the top at the same speed?  (c) Repeat both parts if the car is fully loaded to 1400 kg, which increases friction to 520 N.

38.  A pulley system is used to lift a heavy engine of mass 75.0 kg.  The rope passes around three pulleys such that it is necessary to pull 3.00 m of rope through the system in order to lift the engine 1.00 m.  (a) Ignoring friction, find the amount of force that must be applied to the rope to lift the engine.  (b) Suppose the actual amount of force required is 325 N – find the efficiency of the pulley system.  (c) Find the amount of work done by friction if the engine is lifted 1.00 m.

39.  A certain mechanical jack that is 80.0% efficient is used to lift one side of a car with mass 1500 kg (the jack supports half the weight).  The jack is operated by a crank, the handle of which turns in a radius of 20.0 cm.  Every one turn of the crank lifts the car by 7.5 mm.  (a) Find the amount of force that must be exerted on the handle to lift the car.  (b) How much work must the person do to lift the car 30.0 cm?  (c) How much energy is “wasted” in the process and what becomes of it?

40.  A certain car of mass 1400 kg gets 72 km per gallon (45 mpg) when cruising at a constant 25 m/s over level ground.  The car’s engine/transmission is 35% efficient and runs on gasoline that has 110 MJ energy per gallon.  (a) Find the power input in Watts.  (b) Find the power output of the engine/transmission in Watts and horsepower.  (c) Find the total amount of friction.  (d) Traveling at the same speed, find the number of km/gallon when climbing a slope of 4.0° (assume friction does not change).

1.      a. 346 J
b. −346 J
c. 0

2.      a. 221 J
b. −221 J
c. 442 J

3.      a. 3150 J
b. −3150 J

4.      a. 33 steps
b. 2400 kg

5.      a. 0.15 J
b. 0.15 J
c. 0.15 J

6.

7.      a. −0.75 J
b. −2.1 J
c. −3.2 J

8.      a. 0.025 J
b. −0.079 J

9.      a. −0.250 J
b. 0.500 J
c. −0.250 J
d. 0.960 J

10.  a. 583 kJ
b. 117 m/s
c. 681 N, up
d. 84.4 m

11.  3500 N into the tree

12.  2.7 m/s

13.  a. 80.6 km
b. 69.2 km
c. 52.1 km

14.  a. 414 kJ
b. 2070 N
c. 111 times, 22.1 km

15.

16.  a. 24.4 m/s
b. No – explain!
c. throw 310 gram rock w/ v = 20.0 m/s
throw 155 gram rock w/ v = 31.6 m/s

17.  a. 88.2 J
b. 3.70 m/s
c. 5.42 m/s
d. No – explain!

18.  a. 3.64 m/s, west; 44.1 m/s2, down
b. 4.37 m/s down; 64.5 m/s2, 351.3°

19.  cos−1(2/3) = 48.2°

20.  a.
b.

21.  a. 10.6 m/s
b.

22.  a. −10.9 J
b. 3.62 N
c. 40.9 J; 13.6 N

23.  a. 397 J
b. 235 J
c.

24.  a. 517 m/s
b. altitude 1460 km
c. would escape Ceres’ gravity
and fly off into space at 186 m/s
d. 28 m (92 feet!)

25.  a. −1.33 × 1010 J
b. 6694 m/s
c. 2221 m/s

26.  a. 7641 m/s
b. 1.666 × 1010 J

27.  a. 65.7 m
b. 28.9 m/s
c.

28.  a.
b.

29.  a. 0.785 m/s
b. 0.226 m
c.

30.  a.
b.
c.
d.

31.  a. U = −x2 + x4
b.
c.
d. 0.667 m/s2, left
e. 0.408 m/s
f. 2.83 m/s

32.  a.
b.
c.

33.  49.0 kW

34.  a. 843 kJ
b. −376 kJ
c. 3.1 s

35.  a. 672 W
b. 4.08 s

36.  a. 3700 N
b. 210 hp

37.  a. 17 kW
b. 4.9 km
c. 23 kW; 7.0 km

38.  a. 245 N
b. 75.4%
c. −240 J

39.  a. 55 N
b. 2800 J
c. 550 J

40.  a. 38 kW
b. 13 kW, 18 hp
c. 530 N
d. 26 km/gal