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Circular Motion formulas

Master Circular Motion through 32 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Circular Motion, every formula

32 formulas, typeset and free. Print it, or keep it open beside your practice.

Arc–angle relation
$s = r θ$
Angular velocity
$ω = \frac{d θ}{d t}$
Q1MCQAngular velocity
Angular velocity is defined as:
- A
  $\dfrac{d\theta}{dt}$
- B
  $\dfrac{ds}{dt}$
- C
  $\dfrac{dv}{dt}$
- D
  $r\theta$
Angular acceleration
$α = \frac{d ω}{d t}$
Q1MCQAngular acceleration
Angular acceleration is:
- A
  $\dfrac{d\omega}{dt}$
- B
  $\dfrac{d\theta}{dt}$
- C
  $\omega r$
- D
  $\omega^{2}r$
Speed–angular velocity
$v = r ω$
Centripetal acceleration
$a_{c} = \frac{v ^{2}}{r} = ω^{2} r$
Tangential acceleration
$a_{t} = \frac{d v}{d t} = r α$
Total acceleration
$a = a_{c}^{2} + a_{t}^{2}$
Centripetal force
$F_{c} = \frac{m v ^{2}}{r} = m ω^{2} r$
Time period
$T = \frac{2 π}{ω} = \frac{2 π r}{v}$
Q1MCQTime period
The time period of uniform circular motion is:
- A
  $\dfrac{2\pi r}{v}$
- B
  $\dfrac{v}{2\pi r}$
- C
  $2\pi r v$
- D
  $\dfrac{r}{v}$
Angular kinematics (1)
$ω = ω_{0} + α t$
Angular kinematics (2)
$θ = ω_{0} t + \frac{1}{2} α t^{2}$
Angular kinematics (3)
$ω^{2} = ω_{0}^{2} + 2 α θ$
Conical pendulum tension
$T cos θ = m g, T sin θ = m ω^{2} r$
Q1MCQConical pendulum
For a conical pendulum, the vertical equation is:
- A
  $T\cos\theta=mg$
- B
  $T\sin\theta=mg$
- C
  $T=mg$
- D
  $T\cos\theta=m\omega^{2}r$
Conical pendulum period
$T_{p} = 2 π \frac{L cos θ}{g}$
Q1MCQConical period
The period of a conical pendulum is:
- A
  $2\pi\sqrt{\dfrac{L\cos\theta}{g}}$
- B
  $2\pi\sqrt{\dfrac{L}{g}}$
- C
  $2\pi\sqrt{\dfrac{g}{L}}$
- D
  $2\pi\sqrt{\dfrac{L\sin\theta}{g}}$
Bending of a cyclist
$tan θ = \frac{v ^{2}}{r g}$
Q1MCQCyclist bending
A cyclist rounding a curve bends at angle $\theta$ with:
- A
  $\tan\theta=\dfrac{v^{2}}{rg}$
- B
  $\sin\theta=\dfrac{v^{2}}{rg}$
- C
  $\tan\theta=\dfrac{rg}{v^{2}}$
- D
  $\theta=\dfrac{v^{2}}{rg}$
Banking without friction
$tan θ = \frac{v ^{2}}{r g}$
Q1MCQBanking (no friction)
For a frictionless banked road, the banking angle satisfies:
- A
  $\tan\theta=\dfrac{v^{2}}{rg}$
- B
  $\tan\theta=\dfrac{rg}{v^{2}}$
- C
  $\sin\theta=\dfrac{v^{2}}{rg}$
- D
  $\cos\theta=\dfrac{v^{2}}{rg}$
Banking with friction (max)
$v_{m a x} = \frac{r g ( μ + tan θ )}{1 - μ tan θ}$
Q1MCQBanking (friction)
The maximum safe speed on a banked road with friction is:
- A
  $\sqrt{\dfrac{rg(\mu+\tan\theta)}{1-\mu\tan\theta}}$
- B
  $\sqrt{\mu rg}$
- C
  $\sqrt{rg\tan\theta}$
- D
  $\sqrt{rg}$
Skidding on a level road
$v \leq μ r g$
Q1MCQLevel-road skidding
The maximum speed on a flat curved road is:
- A
  $\sqrt{\mu r g}$
- B
  $\sqrt{rg}$
- C
  $\mu rg$
- D
  $\sqrt{\dfrac{rg}{\mu}}$
Skidding on rotating platform
$ω_{m a x} = \frac{μg}{r}$
Q1MCQRotating platform
The maximum angular speed before an object skids off a rotating platform is:
- A
  $\sqrt{\dfrac{\mu g}{r}}$
- B
  $\sqrt{\mu g r}$
- C
  $\dfrac{\mu g}{r}$
- D
  $\sqrt{\dfrac{g}{r}}$
Concave bridge normal force
$N = m g cos θ + \frac{m v ^{2}}{r}$
Q1MCQConcave bridge
The normal force on a car at the bottom of a concave bridge is:
- A
  $mg\cos\theta+\dfrac{mv^{2}}{r}$
- B
  $mg\cos\theta-\dfrac{mv^{2}}{r}$
- C
  $mg$
- D
  $\dfrac{mv^{2}}{r}$
Convex bridge normal force
$N = m g cos θ - \frac{m v ^{2}}{r}$
Q1MCQConvex bridge
The normal force on a car at the top of a convex bridge is:
- A
  $mg\cos\theta-\dfrac{mv^{2}}{r}$
- B
  $mg\cos\theta+\dfrac{mv^{2}}{r}$
- C
  $mg$
- D
  $\dfrac{mv^{2}}{r}$
Vertical circle: min speed at top
$v_{t o p} = g L$
Q1MCQTop of vertical circle
The minimum speed at the top of a vertical circle of radius $L$ is:
- A
  $\sqrt{gL}$
- B
  $\sqrt{5gL}$
- C
  $\sqrt{2gL}$
- D
  $\sqrt{gL/2}$
Vertical circle: min speed at bottom
$v_{b o tt o m} = 5 g L$
Q1MCQBottom of vertical circle
The minimum speed at the bottom to complete a vertical circle of radius $L$ is:
- A
  $\sqrt{5gL}$
- B
  $\sqrt{gL}$
- C
  $\sqrt{2gL}$
- D
  $\sqrt{3gL}$
Vertical circle: oscillation condition
$u \leq 2 g L$
Q1MCQOscillation condition
A body on a string oscillates (never completes the circle) if the bottom speed satisfies:
- A
  $u\le\sqrt{2gL}$
- B
  $u\ge\sqrt{5gL}$
- C
  $u=\sqrt{gL}$
- D
  $u>\sqrt{2gL}$
Tension at any point (vertical circle)
$T - m g cos θ = \frac{m v ^{2}}{L}$
Q1MCQTension in vertical circle
At angle $\theta$ from the bottom of a vertical circle, the tension satisfies:
- A
  $T-mg\cos\theta=\dfrac{mv^{2}}{L}$
- B
  $T+mg\cos\theta=\dfrac{mv^{2}}{L}$
- C
  $T=mg$
- D
  $T=\dfrac{mv^{2}}{L}$
Tension at bottom (min looping)
$T_{b o tt o m} = 6 m g$
Q1MCQTension at bottom
For minimum looping, the string tension at the lowest point is:
- A
  $6mg$
- B
  $mg$
- C
  $3mg$
- D
  $0$
Death well / rotor minimum speed
$v_{m i n} = \frac{g r}{μ}$
Q1MCQDeath well
The minimum speed for the 'death-well' (rotor) motorcyclist is:
- A
  $\sqrt{\dfrac{gr}{\mu}}$
- B
  $\sqrt{\mu g r}$
- C
  $\sqrt{gr}$
- D
  $\sqrt{\dfrac{\mu g}{r}}$
Centrifugal (pseudo) force
$f = m ω^{2} r (outward)$
Q1MCQCentrifugal force
In the rotating frame, the centrifugal force is:
- A
  $m\omega^{2}r$ outward
- B
  $m\omega^{2}r$ inward
- C
  $mg$
- D
  $0$
Radius of curvature
$R = \frac{v ^{2}}{a _{⊥}}$
Q1MCQRadius of curvature
The radius of curvature of a path is:
- A
  $\dfrac{v^{2}}{a_\perp}$
- B
  $\dfrac{a_\perp}{v^{2}}$
- C
  $v^{2}a_\perp$
- D
  $\dfrac{v}{a_\perp}$
Effect of Earth's rotation
$N = m g - m R ω^{2} cos^{2} λ$
λ = latitude
Q1MCQEarth's rotation
The apparent weight due to Earth's rotation at latitude $\lambda$ is:
- A
  $mg-mR\omega^{2}\cos^{2}\lambda$
- B
  $mg+mR\omega^{2}\cos^{2}\lambda$
- C
  $mg$
- D
  $mR\omega^{2}$
Frequency
$f = \frac{1}{T} = \frac{ω}{2 π}$
Non-uniform: speed changes
$a_{t} \neq = 0 \Rightarrow speed varies$
Q1MCQNon-uniform motion
In non-uniform circular motion, the speed changes because:
- A
  $a_t\neq0$
- B
  $a_c=0$
- C
  $\omega$ is constant
- D
  $r$ changes

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Circular Motion formulas

Circular Motion, every formula

Arc–angle relation

Angular velocity

Angular acceleration

Speed–angular velocity

Centripetal acceleration

Tangential acceleration

Total acceleration

Centripetal force

Time period

Angular kinematics (1)

Angular kinematics (2)

Angular kinematics (3)

Conical pendulum tension

Conical pendulum period

Bending of a cyclist

Banking without friction

Banking with friction (max)

Skidding on a level road

Skidding on rotating platform

Concave bridge normal force

Convex bridge normal force

Vertical circle: min speed at top

Vertical circle: min speed at bottom

Vertical circle: oscillation condition

Tension at any point (vertical circle)

Tension at bottom (min looping)

Death well / rotor minimum speed

Centrifugal (pseudo) force

Radius of curvature

Effect of Earth's rotation

Frequency

Non-uniform: speed changes

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