Circular Motion and Gravitation — Hard Practice Quiz

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

Practice quiz

1. An object undergoes uniform circular motion with radius $r$ and period $T$. Which of the following expressions correctly represents its centripetal acceleration $a_c$?

   • $a_c = \frac{4\pi^2 r}{T^2}$
   • $a_c = \frac{2\pi r}{T^2}$
   • $a_c = \frac{4\pi^2 r^2}{T}$
   • $a_c = \frac{v^2 T}{2\pi r}$

   Answer: $a_c = \frac{4\pi^2 r}{T^2}$

2. A particle of mass $m$ moves in a circular path of radius $r$ with a period $T$. What is the magnitude of the centripetal force acting on the particle?

   • $F_c = \frac{2\pi m r}{T^2}$
   • $F_c = \frac{4\pi^2 m r}{T^2}$
   • $F_c = \frac{m v T}{2\pi r}$
   • $F_c = \frac{m v^2 T}{2\pi r^2}$

   Answer: $F_c = \frac{4\pi^2 m r}{T^2}$

3. An object of mass $m$ is moving in a circular path with speed $v$ and radius $r$, experiencing a centripetal force $F_c$. If the object's speed is doubled to $2v$ and its orbital radius is simultaneously halved to $r/2$, what is the new centripetal force in terms of $F_c$?

   • $2 F_c$
   • $4 F_c$
   • $8 F_c$
   • $16 F_c$

   Answer: $8 F_c$

4. A friction-free banked curve of radius $r$ is designed for vehicles to negotiate it at a specific speed $v$. If a vehicle completes one full traverse of this curve in time $T$, what is the tangent of the banking angle $\theta$?

   • $\tan \theta = \frac{2\pi r}{T^2 g}$
   • $\tan \theta = \frac{4\pi^2 r}{T^2 g}$
   • $\tan \theta = \frac{4\pi^2 r^2}{T^2 g}$
   • $\tan \theta = \frac{v T}{2\pi r g}$

   Answer: $\tan \theta = \frac{4\pi^2 r}{T^2 g}$

5. A satellite is in a circular orbit around Earth with orbital speed $v$. If the orbital radius of the satellite is quadrupled, how does its new orbital speed $v'$ compare to its original speed $v$?

   • $v' = 4v$
   • $v' = 2v$
   • $v' = \frac{v}{2}$
   • $v' = \frac{v}{4}$

   Answer: $v' = \frac{v}{2}$

6. A satellite of mass $m$ orbits Earth in a circular path of radius $r$ with speed $v$. If an identical satellite with mass $2m$ were placed in the same orbit (same radius $r$), what would be its orbital speed $v'$?

   • $v' = \sqrt{2} v$
   • $v' = 2v$
   • $v' = \frac{v}{\sqrt{2}}$
   • $v' = v$

   Answer: $v' = v$

7. A satellite orbits Earth with a period $T$. If a new satellite is placed in an orbit such that its period is $8T$, by what factor must its orbital radius have changed compared to the original satellite?

   • $2$
   • $4$
   • $8$
   • $16$

   Answer: $4$

8. An object of mass $m$ moves in a circular path of radius $R$ with speed $V$, experiencing a centripetal force $F$. If the speed is increased to $3V$ and the radius is simultaneously increased to $2R$, what is the new centripetal force $F'$ in terms of $F$?

   • $\frac{3}{2} F$
   • $\frac{9}{4} F$
   • $\frac{9}{2} F$
   • $9 F$

   Answer: $\frac{9}{2} F$

9. A small object of mass $m$ is whirled in a horizontal circle of radius $r$. If it completes $N$ revolutions in a total time $t$, what is the magnitude of the centripetal force acting on the object?

   • $F_c = \frac{2\pi m r N}{t^2}$
   • $F_c = \frac{4\pi^2 m r N^2}{t^2}$
   • $F_c = \frac{m r N^2}{4\pi^2 t^2}$
   • $F_c = \frac{m v^2 N}{t}$

   Answer: $F_c = \frac{4\pi^2 m r N^2}{t^2}$

10. A banked curve is designed for a specific speed $v_0$ such that no friction is required. If a car of mass $m$ attempts to take this curve at a speed of $2v_0$, what additional inward force (e.g., friction) is required to keep the car from sliding outwards?

    • $\frac{mv_0^2}{r}$
    • $\frac{2mv_0^2}{r}$
    • $\frac{3mv_0^2}{r}$
    • $\frac{4mv_0^2}{r}$

    Answer: $\frac{3mv_0^2}{r}$