### Vertical circular motion

Common examples include:

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

We use DID TASC to analyze all these problems.

The Water Bucket

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):   ΣFr = N - mg = mar = m v2/r      N = m v2/r + mg

At the top (y=down)      ΣFr = N + mg = mar = 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:   ΣFr = N = mar = 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).

Car on a Hilly Road

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.