AP Physics Assignment – Dynamics:  Forces and Newton’s Laws of Motion

Reading    Physics for Scientists and Engineers – Chapter 5 and pp. 148 – 153

Objectives/HW

 The student will be able to: HW: 1 State Newton’s 1st and 2nd Laws of Motion and apply these laws to physical situations in order to determine what forces act on an object and to explain the object’s resulting behavior.  Define and apply the concept of inertia and inertial frame of reference. 1 – 7 2 Recognize and state the proper SI unit of force and give its equivalence in fundamental units and use the relation Fnet = ma to solve problems. 8 – 10 3 Recognize the difference between weight and mass and convert from one to the other. 11 – 14 4 State and utilize Newton’s 3rd Law to solve related problems. 15 – 18 5 Understand and utilize the concept of the normal force to solve related problems. 19, 20 6 Define and apply the concepts of compression and tension and use the method of sections to solve for these. 21 – 26 7 Solve force problems involving pulleys, including those involving multiple objects and systems of equations (such as Atwood’s machine). 27 – 31 8 Understand and utilize the relation between friction force, normal force, and coefficient of friction for both cases:  static and kinetic. 32 – 37 9 Solve problems involving air resistance in which friction is assumed directly proportional to velocity; define and apply the concept of terminal velocity. 38 – 39 10 Apply the concept of force components to objects on an incline and solve related problems. 40 – 44

Homework Problems

1.      A classic “parlor” trick is to grab a tablecloth and pull it out from under the plates, silverware, and tumblers of a full dinner setting.  If done properly, all of the place settings hardly move as the tablecloth is pulled out.  Explain how this trick “works” in light of Newton’s laws of motion.

2.      Describe the type of path that must be followed by an object moving in the absence of force.  Explain.

3.      Must an object always move in the direction of the net force acting upon it?  Explain and give examples to support your answer.

4.      A cart on wheels is loaded with books.  In order to move the cart you must exert a relatively great force to start it moving but then a much smaller force to keep it moving.  Use Newton’s laws to explain.

5.      In order for an object to move at greater and greater speeds does it require a greater and greater amount of force?  Explain your answer using the laws of motion.

6.      Consider what happens when a car collides with a tree and the passenger in the front seat is not wearing a seatbelt.  Refer to Newton’s laws of motion in order to explain each response:  (a) What causes the person to be “thrown” into the dash?  (b) In the brief time that the person’s body is moving between the seat and the dash, is the person accelerating?  (c) Repeat part (b) using the car’s body as a frame of reference.  (d) Explain why the car is not an inertial reference frame.

7.      In an old pickup truck there may be a window directly behind the driver’s head – often with no headrest.  If such a truck is at rest and is hit from behind by another car, the driver’s head is likely to impact the window behind it.  (a) Explain why this happens.  (b) Would the result be different if the truck was initially in motion instead of at rest?  Explain.

8.      A Mini Cooper has a mass of 1290 kg and can go from 0 to 26.8 m/s in 6.2 s.  What average amount of net force must act on the car in order for this to occur?

9.      In reality the net force on a car varies as it accelerates.  The net force on the Mini from the previous problem can be modeled by F = 6579 − 387t, where F is in newtons and t is in seconds, when it is accelerating maximally from rest.  (a) Find the maximum amount of acceleration.  (b) Find the magnitude of acceleration and speed at t = 2.00 s.  (c) Determine the time for it to go from zero to 20.0 m/s.  (d) Determine the time for it to go from 20.0 m/s to 40.0 m/s.

10.  In a physics experiment an object is observed to accelerate 2.5 m/s2, 0.0° when under the influence of two forces:  F1 = 5.0 N, 0.0° and F2 = 1.5 N, 180.0°.  Determine the mass of the object.

11.  A toolbox of mass 3.2 kg is lowered by rope from the roof to the ground.  Find the acceleration of the toolbox when the force of the rope is:  (a) 40.0 N and (b) 20.0 N.

12.  A package that weighs 50.0 N is lifted by a person and accelerates upward 2.50 m/s2.  What force must the person exert for this to happen?

13.  David Blaine performed a stunt in which he fell from atop a 27 m pole onto a stack of cardboard boxes 3.7 m high.  Assuming Blaine had a mass of 85 kg, estimate the minimum force that the boxes exerted on Blaine as he crashed through the stack.

14.  A girl throws a 175 gram softball by exerting upon it a force of 10.0 N, 45.0°.  (a) Determine the resulting acceleration of the softball.  (b) Repeat for the same amount of force applied by the girl in a downward direction.

15.  Identify and describe at least one action-reaction pair of forces in each of the following scenarios:  (a) A person climbs a ladder.  (b) A propeller-driven airplane flies.  (c) A soccer ball is kicked.  (d) A car accelerates from a stoplight.

16.  In order for a 70.0 kg sprinter to accelerate 4.00 m/s2 eastward, what force must he exert on the blocks at the starting line?  Explain how Newton’s 3rd Law relates to this.

17.  An apple of mass 444 g breaks loose of a tree and falls for 1.11 s before being caught by a person.  The person’s hand is moved downward 9.99 cm by the apple as it is caught.  (a) Describe the action-reaction pairs of forces for the apple and the earth and for the apple and the hand.  (b) Find the amount that the earth (m = 5.974 × 1024 kg) moves upwards while the apple is falling.  (c) Find the average amount of force that the apple exerts on the person’s hand.

18.  Mules are smart but stubborn.  Once upon a time a particularly smart and particularly stubborn mule refused to pull its owner’s cart and gave the following argument:  “I refuse to pull the cart because it is impossible to do so according to Newton’s laws of motion.  According to the 2nd Law it is necessary to have a net force in order for the cart to accelerate.  According to the 3rd Law no matter how hard I pull the cart forward, the cart will pull an equal amount backward and therefore the net force will be zero and the cart will not move.  Even if I could pull a million newtons forward, the cart would pull a million newtons backward and so I refuse to even try!”  Where is the flaw in the mule’s argument?

19.  A basketball that weighs 5.84 N is dribbled on the floor.  In the process of bouncing the ball has an acceleration of 80.0 m/s2 upward.  (a) Determine the normal force the floor exerts on the ball.  (b) Determine the normal force that the ball exerts on the floor.

20.  A metal box is tilted at an angle of 30.0° as shown below.  Inside the box rests a bowling ball of mass 5.00 kg.  Determine the normal force at each point of contact.

21.  A traffic signal of mass 25.0 kg is suspended by two cables as shown in the diagram below.  Solve for the tension in each cable each of the following scenarios:  (a) θ1 = θ2 = 20.0°, and (b) θ1 = 20.0°, θ2 = 30.0°.

22.  A pair of fuzzy dice hangs from the ceiling of a car that accelerates forward at 3.00 m/s2.  Assuming the dice have the same acceleration as the car, what is the angle θ as shown in the diagram below?

23.  Suppose there is a stack of ten books on the floor.  Each book has a mass of 1.40 kg.  (a) Determine the force that the eighth book exerts on the ninth book.  (count up from the bottom)  (b) Determine the force that the second book exerts on the first.  (c) Find the normal force the first book exerts on the floor if a boy with mass 60.0 kg jumps off the top of the stack with an acceleration of 3.00 m/s2, upward.

24.  A man of mass 95.00 kg is hanging from the end of a rope of mass 2.50 kg.  The other end of the rope is attached to a helicopter.  If the helicopter rises such that the man accelerates upward at 4.000 m/s2 find the tension in the rope at its:  (a) bottom, (b) middle, and (c) top.

25.  Consider a man standing in an elevator.  The man has a total mass of 70.0 kg.  Each of his legs has a mass of 12.0 kg.  (a) If the elevator is at rest determine the normal force on each foot and the amount of vertical force at the joint of each femur and pelvis.  (b) Repeat with the elevator accelerating upward at 2.00 m/s2.  (c) Repeat with the elevator accelerating downward at 1.5 m/s2.

26.  Two balls of mass m1 and m2 are connected by a string and lifted by an upward force F as shown below.  Derive and simplify an expression for the tension T in the connecting string in terms of relevant variables and constants.  The result should allow for the possibility of acceleration.

27.  Two objects are connected by a cord of negligible mass that passes over a massless, frictionless pulley as shown below.  When the two objects are released, the object with mass 700.0 gram accelerates downward at 3.00 m/s2.  (a) Determine the mass of the other object.  (b) Find the tension in the cord as the objects move.

28.  A window cleaner of mass 75.0 kg lifts himself and his 25.0 kg platform by the rope and pulley system shown in the diagram below.  (a) In order to go up at a constant velocity what force must he exert on the rope?  (b) Determine the amount of normal force on his feet as he rises at constant velocity.  (c) Find the acceleration if the man pulls 525 N on the rope.

29.  Two objects, masses 6.0 kg and 8.0 kg, are connected by a lightweight cord that passes over a massless, frictionless pulley as shown in the diagram.  An upward force of 250.0 N acts on the axle of the pulley, causing the entire system to move upward (the pulley is still free to rotate).  Determine the acceleration of each mass and the acceleration of the pulley.

30.  A truck of mass 2250 kg is used to pull a 330 kg trunk out of an old quarry as shown below.  There is 550 N of rolling resistance for the truck.  Ignore mass and friction of the pulley.  (a) If the trunk is accelerated upward at 2.00 m/s2 what is the minimum amount of horizontal force the truck’s drive wheels exert on the ground?  (b) For the same conditions what is the tension in the towrope?

31.  Suppose the truck in the previous problem pops out of gear and is momentarily in neutral such that it is pulled back by the trunk.  (a) Find the resulting acceleration.  (b) Find the tension in the rope.

32.  Two boys stand on the surface of a frozen lake where μs = 0.15 and μk = 0.10.  The 30.0 kg boy shoves the 40.0 kg boy with a horizontal force of 50.0 N, 0.0°.  Find the resulting acceleration of each boy.

33.  A man of mass 80.0 kg pushes horizontally on a large crate as he moves it across a level floor where μk = 0.30.  When the man’s feet push against the floor 300.0 N to the left, the man and the crate both accelerate 1.50 m/s2 to the right.  (a) Find the force that the man exerts on the crate.  (b) Find the mass of the crate.

34.  A block of mass m is pulled across a level surface by a rope that makes an angle θ with the horizontal.  The coefficient of friction is μ.  (a) Determine the amount of force F required to slide the block at a constant velocity.  (b) Determine the optimum angle at which to pull on the block (so that the required force is minimized).  (c) If the force of the rope is 15.0 N acting on a block of mass 2.00 kg where μ = 0.35, what is the maximum acceleration possible?

35.  A man pushes 100.0 N on the handle of a 40.0 kg lawnmower.  The applied force is in the same direction as the handle, which is tilted 35.0° relative to horizontal.  The coefficient of kinetic friction is 0.10.  Find the amount of acceleration.

36.  A large crate sits in the back of a pickup truck.  The coefficients of friction are: μs = 0.60 and μk = 0.40.  (a) Find the maximum forward acceleration of the truck at which the crate will not slide backward.  (b) Suppose the truck decelerates at 8.0 m/s2 and the crate slides 1.5 m forward before hitting the front end of the bed.  Find the impact speed (relative to the truck).

37.  Friction or “retarding” forces on a car comes in three types:  rolling resistance, air resistance, and internal friction associated with the engine and transmission.  On a particular small car of mass 900 kg, rolling resistance can be modeled with a coefficient μ = 0.024 and the equation:  Ff = μFN.  If this car can accelerate from 0 to 25 m/s in 9.0 s with a single 100.0 kg driver, find the time for the same change in speed with three additional passengers of the same mass.

38.  A rock of mass 0.400 kg is released from the surface and sinks in the ocean.  As the rock descends it is acted upon by three forces:  gravity, buoyancy, and drag.  The buoyancy is an upward force equal to half its weight.  Drag from the water can be modeled by F = kv, where k = 0.650 kg/s.  (a) Determine the terminal speed of the sinking rock.  (b) Determine its depth, speed, and acceleration 1.50 seconds after it is released.  (c) At what depth will it be at 99.0% of its terminal speed?

39.  A certain plastic ball of mass 0.150 kg has a terminal speed of 20.0 m/s when falling through air.  This ball is launched upward at 16.0 m/s.  Assume that air resistance is proportional to speed.  (a) Determine the time it spends rising and the maximum height that it will reach.  (b) Make careful sketches of velocity vs. time and acceleration vs. time.

40.  Consider two kids on skateboards coasting down a hill.  (a) Ignoring air resistance, derive an expression for the acceleration, a, as a function of:  angle of incline, θ, coefficient of friction, μ, and g.  (b) According to the result, the mass of the kid should not matter.  However, in reality, the more massive kid will usually have a greater acceleration.  Explain!

41.  A small box is at the bottom of a ramp tilted at an angle of 40.0° above horizontal.  The box is given a push and it then slides up the ramp 2.0 seconds before sliding back down.  The coefficient of sliding friction is 0.15.  Find the time for it to slide back to the point at which it was released.

42.  A block is set on an adjustable ramp and the angle of incline is increased slowly until the block is observed to start sliding.  (a) Solve for the coefficient of static friction, μs, in terms of the maximum angle of incline, θ, at which the block can remain at rest on the ramp.  (b) Considering the fact that μs is almost always less than or equal to one, at what amount of incline would there be too little friction to prevent virtually anything from sliding?

43.  A block of mass 2.00 kg rests on a ramp tilted at 25.0°.  Coefficients of friction for the block/ramp:  μs = 0.300, μk = 0.100.  A string passing over a high quality pulley connects the block with a bucket as shown below.  The bucket contains sand but a hole is punched in the bottom and it starts to run out.  When there is a certain amount of sand left in the bucket the block starts to move. (a) What mass of sand is left in the bucket when it starts to move?  (b) Find the acceleration rate of the block/bucket system shortly after motion begins.

44.  As shown in the diagram below, two blocks are connected by a lightweight cord that passes over a frictionless, massless pulley.  The bottom block has a mass of 2.00 kg and the top block has mass 0.500 kg.  The coefficient of kinetic friction at each sliding surface is equal to 0.10.  The two blocks are release from rest.  Determine the acceleration of each block.

1.

2.

3.

4.

5.

6.      a. thru d.

7.      a. b.

8.      5600 N

9.      a. 5.10 m/s2
b. 4.50 m/s2, 9.60 m/s
c. 4.52 s
d. 7.75 s

10.  1.4 kg

11.  a. 2.7 m/s2, up
b. 3.6 m/s2, down

12.  62.8 N, up

13.  6.1 kN, upward

14.  a. 50.7 m/s2, 37.1°
b. 66.9 m/s2, 270.0°

15.  a. thru d.

16.  280 N, westward

17.  a.
b. 4.49 × 10−25 m
c. 267 N, downward

18.

19.  a. 53.5 N, up
b. 53.5 N, down

20.  24.5 N, 30.0° and 42.4 N, 120.0°

21.  a. 358 N in each
b. 277 N and 301 N

22.  17.0°

23.  a. 27.4 N, up
b. 123 N, down
c. 905 N, down

24.  a. 1311 N
b. 1328 N
c. 1346 N

25.  a. 343 N ea. foot, 225 N ea. joint
b. 413 N ea. foot, 271 N ea. joint
c. 291 N ea. foot, 191 N ea. joint

26.

27.  a. 372 g
b. 4.76 N

28.  a. 490 N, down
b. 245 N
c. 0.700 m/s2, up

29.  11 m/s2, up (6 kg)
5.8 m/s2, up (8 kg)
8.4 m/s2, up (pulley)

30.  a. 8.9 kN
b. 3.9 kN

31.  a. 1.0 m/s2, back
b. 2.9 kN

32.  0.69 m/s2, 180° (smaller boy)
the larger boy does not move

33.  a. 180 N, right
b. 41 kg

34.  a.
b.
c. 4.5 m/s2, 0°

35.  0.92 m/s2

36.  a. 5.9 m/s2, forward
b. 3.5 m/s

37.  12 s

38.  a. 3.02 m/s
b. 2.84 m, 2.75 m/s, 0.428 m/s2, down
c. 6.71 m

39.  a. 1.20 s, 8.66 m
b.

40.  a.
b.

41.  2.4 s

42.  a. μs = tan θ
b. 45°

43.  a. 0.301 kg
b. 1.54 m/s2

44.  big block:  1.75 m/s2 330.0°
small block:  1.75 m/s2 150.0°