Common examples include:

• roller coasters
• water buckets
• cars traveling on hilly roads

We use DID TASC to analyze all these problems.

Step 1: Draw a diagram and coordinate system. Step 2: Isolate system.

Step 3: Draw all forces. Step 4: Take components.

At rest:

• Constant downward force of gravity
• Constant upward normal force

Crucial feature of uniform circular motion in the vertical plane:

• Constant downward force of gravity
• Non-Constant upward normal force

Step 5: Apply Newton's second law.

At the bottom (y=up):   ΣF_r = N - mg = ma_r = m v²/r      N = m v²/r + mg

At the top (y=down)      ΣF_r = N + mg = ma_r = m v²/r      N = m v²/r - mg

Step 6: Solve. How slow can you go at the top and not fall off?

The limit is N =0! The corresponding speed is:  v² = gr

Halfway up or down:   ΣF_r = N = ma_r = m v²/r      N = m v²/r

At these points, mg provides the tangential acceleration that slows the object down (on the way up) or speeds it up (on the way down).

What is the normal force on a car when it moves with speed v at the bottom of a circular valley of radius r?

1. N = mg
2. N = mv²/r
3. N = mg + mv²/r
4. N = mg - mv²/r
5. N = mv²/r - mg

At the bottom:

N

=

mg +

m v2 r

What is the normal force on a car when it moves with speed v at the top of a circular hill of radius r?

1. N = mg
2. N = mv²/r
3. N = mg + mv²/r
4. N = mg - mv²/r
5. N = mv²/r - mg

At the top:

N

=

mg -

m v2 r

The normal force equals your apparent weight. This is why you feel heavier at the bottom and lighter at the top.

A key difference between the car and water bucket examples: The normal is always up for the car. For the bucket, the normal force points from the bottom of the bucket toward the top of the bucket.