GRE Physics Companion: Dynamics of Circular Motion

This companion collects fast solution patterns for circular-motion questions. The governing idea is always

(1)

Do not add a separate centripetal force. Instead, identify which real forces have inward components.

1 Fast triage

The most useful standard forms are

(2)

(3)

(4)

for a frictionless banked turn, and

(5)

for local contact at the top of an inside vertical loop.

2 Scaling shortcuts

From a_n = v²∕R:

• doubling v multiplies a_n and the required inward force by 4;
• doubling R at fixed v halves them;
• at fixed angular speed, a_n = ω²R grows linearly with radius.

3 Worked GRE example 1: flat curve scaling

A CAR can just round a level curve of radius R at speed v without slipping. On the same surface, what is the maximum speed on a curve of radius 4R?

The flat-curve limit is

(6)

With the same surface, μ_s is unchanged. Therefore

(7)

Hence

(8)

The speed scales with the square root of radius, not directly with radius.

4 Worked GRE example 2: top of a loop

A cart moves on the inside of a vertical loop. At the top its speed is exactly . What is the Normal force there?

At the top, inward is downward:

(9)

Since v² = 2gR,

(10)

Therefore

(11)

The common trap is to set N = mv²∕R and forget that gravity also contributes inward.

5 GRE practice questions

1. A particle’s speed around a fixed-radius circle is tripled. By what factor does the required inward net force change?
2. A car turns on a level road at speed v. The coefficient of static friction is doubled while the curve radius is unchanged. By what factor does the maximum possible no-slip speed change?
3. A frictionless banked curve is designed for speed v and radius R. If the radius is quadrupled while the bank angle is unchanged, what is the new design speed?
4. A rider moves through the bottom of a circular dip. Which is larger there: the normal force or the rider’s Weight? Assume nonzero speed.
5. A car moves over the top of a convex hill. As its speed increases while R remains fixed, what happens to the normal force?
6. A cart is at the top of an inside vertical loop with speed v = . What is the normal force?
7. A conical pendulum is held at a larger angle from the vertical while its string length remains fixed. Does its angular speed increase, decrease, or remain unchanged?

6 Answers

1. Factor of 9.
2. Factor of .
3. Factor of 2.
4. The normal force is larger than the weight because N − mg = mv²∕R > 0.
5. It decreases according to N = mg − mv²∕R and reaches zero at contact loss.
6. Zero. Gravity alone supplies the required inward force.
7. It increases, because ω² = g∕(L cos 𝜃) and cos 𝜃 decreases.

7 Final checklist

• Choose the inward direction before writing signs.
• Draw only real forces.
• Use ∑ F_n = mv²∕R for the normal direction.
• Use ∑ F_t = mdv∕dt only when the speed is changing.
• Remember that N cannot be negative for passive contact.
• Check the v² scaling before accepting an answer choice.

References