Circular Motion and Gravitation — Practice Quiz

A Physics cheat sheet for Circular Motion and Gravitation — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Uniform Circular Motion (Speed): The speed \(v\) of an object moving in a circle of radius \(r\) with period \(T\).

$$v = \frac{2\pi r}{T}$$

Centripetal Acceleration: The acceleration \(a_c\) directed toward the center of the circle for an object in uniform circular motion.

$$a_c = \frac{v^2}{r}$$

Centripetal Force: The net force \(F_c\) required to keep an object of mass \(m\) moving in a circular path.

$$F_c = \frac{mv^2}{r}$$

Banked Curves: The angle \(\theta\) at which a friction-free curve of radius \(r\) must be banked for a vehicle to negotiate it at speed \(v\).

$$\tan \theta = \frac{v^2}{rg}$$

Satellite Speed: The speed \(v\) of a satellite in a circular orbit of radius \(r\) around the Earth (mass \(M_E\)).

$$v = \sqrt{\frac{G M_E}{r}}$$

Where: \(G\) = gravitational constant

Satellite Period: The time \(T\) required for a satellite to complete one orbit around the Earth.

$$T = \frac{2\pi r^{3/2}}{\sqrt{G M_E}}$$

Practice quiz

  1. A car travels around a circular track with a radius of $50 \text{ m}$ and completes one lap in $10 \text{ s}$. What is the speed of the car?

    • A) $10\pi \text{ m/s}$
    • B) $5\pi \text{ m/s}$
    • C) $20\pi \text{ m/s}$
    • D) $50\pi \text{ m/s}$

    Answer: A) $10\pi \text{ m/s}$

  2. An object moves in a circle with a speed of $10 \text{ m/s}$ and a radius of $5 \text{ m}$. What is its centripetal acceleration?

    • A) $20 \text{ m/s}^2$
    • B) $10 \text{ m/s}^2$
    • C) $50 \text{ m/s}^2$
    • D) $2 \text{ m/s}^2$

    Answer: A) $20 \text{ m/s}^2$

  3. A $2 \text{ kg}$ mass is swung in a horizontal circle of radius $0.5 \text{ m}$ at a speed of $4 \text{ m/s}$. What is the centripetal force acting on the mass?

    • A) $64 \text{ N}$
    • B) $32 \text{ N}$
    • C) $16 \text{ N}$
    • D) $8 \text{ N}$

    Answer: A) $64 \text{ N}$

  4. A particle moves in uniform circular motion with a radius of $2 \text{ m}$ and a period of $4\pi \text{ s}$. What is its centripetal acceleration?

    • A) $0.5 \text{ m/s}^2$
    • B) $1 \text{ m/s}^2$
    • C) $2 \text{ m/s}^2$
    • D) $4 \text{ m/s}^2$

    Answer: A) $0.5 \text{ m/s}^2$

  5. A $0.1 \text{ kg}$ object is swung in a horizontal circle of radius $1 \text{ m}$ with a period of $2 \text{ s}$. What is the centripetal force required?

    • A) $0.1 \pi^2 \text{ N}$
    • B) $0.2 \pi^2 \text{ N}$
    • C) $0.05 \pi^2 \text{ N}$
    • D) $0.1 \pi \text{ N}$

    Answer: A) $0.1 \pi^2 \text{ N}$

  6. A frictionless curve of radius $100 \text{ m}$ is designed for cars to negotiate it safely at a speed of $20 \text{ m/s}$. What is the required banking angle? (Use $g = 9.8 \text{ m/s}^2$)

    • A) $\arctan(\frac{20}{49})$
    • B) $\arctan(\frac{10}{49})$
    • C) $\arctan(\frac{40}{49})$
    • D) $\arctan(\frac{5}{49})$

    Answer: A) $\arctan(\frac{20}{49})$

  7. A satellite orbits Earth at a radius $r$. If the gravitational constant is $G$ and Earth's mass is $M_E$, what is the satellite's orbital speed?

    • A) $\sqrt{\frac{G M_E}{r}}$
    • B) $\frac{G M_E}{r}$
    • C) $\sqrt{\frac{G M_E}{r^2}}$
    • D) $\frac{G M_E}{r^2}$

    Answer: A) $\sqrt{\frac{G M_E}{r}}$

  8. A satellite orbits Earth at a radius $r$. If the gravitational constant is $G$ and Earth's mass is $M_E$, what is the satellite's orbital period?

    • A) $\frac{2\pi r^{3/2}}{\sqrt{G M_E}}$
    • B) $\frac{2\pi r}{\sqrt{G M_E}}$
    • C) $\frac{2\pi \sqrt{r}}{\sqrt{G M_E}}$
    • D) $\frac{2\pi r^2}{\sqrt{G M_E}}$

    Answer: A) $\frac{2\pi r^{3/2}}{\sqrt{G M_E}}$

  9. If an object's speed in uniform circular motion is doubled and its orbital radius is halved, how does the centripetal force acting on it change?

    • A) It increases by a factor of $8$.
    • B) It increases by a factor of $4$.
    • C) It remains the same.
    • D) It decreases by a factor of $2$.

    Answer: A) It increases by a factor of $8$.

  10. Consider a satellite in a circular orbit around Earth. If the satellite's orbital radius increases, what happens to its orbital speed and orbital period?

    • A) Speed decreases, period increases.
    • B) Speed increases, period decreases.
    • C) Both speed and period increase.
    • D) Both speed and period decrease.

    Answer: A) Speed decreases, period increases.

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