"The Elevator Problem"

The "Elevator Problem" is a classic problem in physics. The situation is this:

"You are standing on a bathroom scale in an elevator. You are holding an apple. (Yes, people are staring at you...) You weigh 500 Newtons, so your mass is about 50 kg."

This assignment is a step-by-step analysis of the elevator problem. A good deal of the work has been done for you - in which case it is your job to study the answers given and use them as a pattern and guide for the answers that you supply. Answers are available.

Part A: Elevator Is At Rest.

You have just boarded the elevator, so it (with you inside) is at rest...

Question 1: What does the scale read?

Answer:

There are 2 forces acting on you. (See the diagram at right.) The Earth pulls down on you with the force we call your weight (= mg) of 500 Newtons.

Since the elevator is at rest, your acceleration is 0 m/s². Since your acceleration is 0 m/s², Newton's First Law says the net force on you must be 0 Newtons.

Since the net force on you is 0 Newtons, the upward forces and downward forces on you must balance exactly. Therefore the scale must push on you with a force of (1a) _______ Newtons, and the scale must read (1b) _______ Newtons.

Question 2: If you let go of the apple, what does it do?

Answer:

The apple would be in free fall, so its acceleration relative to the earth is 10 m/s² downward. Since you are at rest relative to the earth, the apple's acceleration relative to you would be10 m/s² also, so the apple would appear to fall just as it does anywhere else on the earth.

Part B: The Elevator Accelerates Upward.

The elevator, (with you inside) begins to accelerate upward from rest at 2 m/s².

Question 3: What will the scale read now?

Answer:

There are 2 forces acting on you. (Complete the diagram.) Your weight pulls down with a force of 500 Newtons. The scale pushes up with a force of (3a) ______ (see below) Newtons.

Since your acceleration is 2 m/s² upward, Newton's Second Law says that there must be a net force pushing you upward, and the net force has a magnitude F_net = ma. So the net force on you,

F_net = (50 kg)(2 m/s²) = 100 Newtons (upward).

Since the net force on you is100 Newtons, the upward forces and downward forces on you must cancel to leave a 100 Newton upward force. Therefore the scale must push on you with a force of (3b) _______ Newtons, and the scale must read (3c) _______ Newtons as the elevator accelerates upward.

Question 4: If you let go of the apple now, what does it do?

Answer:

The apple would be in free fall, so its acceleration relative to the earth is 10 m/s² downward. Since you are accelerating at 2 m/s² upward relative to the earth, the apple's acceleration relative to you would be 10 m/s² + 2 m/s² = 12 m/s², so the apple would appear to fall faster inside the elevator than it does in "normal" free fall on the earth.

To the occupants of the upwardly accelerating elevator, it appears that gravity is stronger, since they seem to weigh more (why?) and objects fall faster than "normal."

Part C: The Elevator Moves Up With Constant Velocity.

The elevator (and you) accelerated for 5 seconds, so it is moving upward with a velocity of 10 m/s. It now moves with this constant upward velocity of 10 m/s.

Question 5: What does the scale read now?

Answer:

There are 2 forces acting on you. (Complete the diagram.) Your weight pulls down with a force of 500 Newtons. The scale pushes up with a force of (5a) ________ (see below) Newtons.

Since the elevator is moving with constant velocity, your acceleration is (5b) ____ m/s². Since your acceleration is (5c) ____ m/s², Newton's First Law says the net force on you is (5d) _____ Newtons.

Since the net force on you is (5e) ____ Newtons, the scale must push on you with a force of (5f) _______ Newtons, and the scale must read (5g) _______ Newtons.

Question 6: If you let go of the apple, what does it do?

Answer:

Part D: The Elevator Slows Down (While Going Up)

The elevator, (with you inside) begins to slow down as it approaches its destination. Its acceleration (or deceleration) is 2 m/s² downward.

Question 7: What does the scale read now?

Answer:

There are 2 forces acting on you. (Complete the diagram.) The Earth (your weight) pulls down with a force of (7a) _____ Newtons. The scale pushes up with a force of (7b)______ (see below) Newtons.

Since your acceleration is 2 m/s² downward, Newton's Second Law says that there must be a net force pulling you downward, and the net force has a magnitude F_net = ma. So the net force on you,

F_net = ((7c) _____ kg)((7d) _____ m/s²) = (7e) ______ Newtons .

Since the net force on you is (7f) _____ Newtons downward, the upward forces and downward forces on you must cancel out to leave a (7g) _____ Newton downward force . Therefore the scale must push on you with a force of (7h) _______ Newtons, and the scale must read (7i) _______ Newtons as the elevator accelerates downward.

Question #8: If you let go of the apple now, what does it do?

Answer:

To the occupants of the downwardly accelerating elevator, it appears that gravity is weaker, since they seem to weigh less (why?) and objects fall more slowly than "normal."

Part E: The Elevator Speeds Up (While Going Down)

The elevator, (with you inside) reaches its floor, stops for a while, and then begins to accelerate downward. Its acceleration is 2 m/s² downward.

Question 9: What does the scale read now?

Answer:

There are 2 forces acting on you. (Complete the diagram.) The Earth (your weight) pulls down with a force of (9a) _____ Newtons. The scale pushes up with a force of (9b)______ (see below) Newtons.

Since your acceleration is 2 m/s² downward, Newton's Second Law says that there must be a net force pulling you downward, and the net force has a magnitude F_net = ma. So the net force on you,

F_net = ((9c) _____ kg)((9d) _____ m/s²) = (9e) ______ Newtons .

Since the net force on you is (9f) _____ Newtons downward, the upward forces and downward forces on you must cancel out to leave a (9g) _____ Newton downward force. Therefore the scale must push on you with a force of (9h) _______ Newtons, and the scale must read (9i) _______ Newtons as the elevator accelerates downward.

Question #10: If you let go of the apple now, what does it do?

Answer:

To the occupants of the downwardly accelerating elevator, it appears that gravity is weaker, since they seem to weigh less (why?) and objects fall more slowly than "normal."

Part F: The Elevator Moves Down With Constant Velocity.

The elevator (and you) accelerated for 5 seconds, so it is moving downward with a velocity of 10 m/s. It now moves with this constant downward velocity of 10 m/s.

Question 11: What does the scale read now?

Answer:

There are 2 forces acting on you. (Complete the diagram.) Your weight pulls down with a force of 500 Newtons. The scale pushes up with a force of (11a) ________ (see below) Newtons.

Since the elevator is moving with constant velocity, your acceleration is (11b) ____ m/s². Since your acceleration is (11c) ____ m/s², Newton's First Law says the net force on you is (11d) _____ Newtons.

Since the net force on you is (11e) ____ Newtons, the scale must push on you with a force of (5f) _______ Newtons, and the scale must read (11g) _______ Newtons.

Question 12: If you let go of the apple, what does it do?

Answer:

Part G: Oh, No!

The elevator cable snaps, and the elevator (with you inside!) begins to fall! Perhaps you have time for one last Physics observation!

Question 13: What does the scale read as the elevator falls?

Answer:

There are 2 forces acting on you. (Complete the diagram.) The Earth (your weight) pulls down with a force of (13a) _____ Newtons. The scale pushes up with a force of (13b) _____ (see below) Newtons.

Since the elevator and you are in free fall, your acceleration is (13c) ____ m/s² downward. Newton's Second Law says that there must be a net force pulling you downward, and the net force has a magnitude F_net = ma. So the net force on you,

F_net = ma = ((13d) _____ kg)((13e) _____ m/s²) = (13f) _____ Newtons.

Since the net force on you = (13g) _____ Newtons downward, the upward forces and downward forces on you must cancel out to leave a (13h) _____ Newton downward force . Therefore the scale must push on you with a force of (13i) _______ Newtons, and the scale must read (13j) _______ Newtons as the elevator accelerates downward.

Question #14: If you let go of the apple now, what does it do?

Answer:

Question #15: "Weightlessness"

Weightlessness is a phenomenon that we most often associate with astronauts in space, but it is not necessary to be floating in space to be "weightless." Describe what you could do, right here and now, to be "weightless." (Hint: An elevator is not necessary!)

Answers are available (really, no tricks this time - promise!).