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Circular Motion formula sheet

Every formula in the Circular Motion chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 35 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Angular variables and centripetal acceleration

Angular variables
11× asked, last 2025
θ=sr,ω=dθdt=2πT=2πf,v=ωr\theta=\frac sr,\qquad \omega=\frac{d\theta}{dt}=\frac{2\pi}{T}=2\pi f,\qquad v=\omega r
θ\theta in radians. NN rpm =2πN/60=2\pi N/60 rad/s. Every point of a rigid body has the same ω\omega.
Angular velocity as a vector
v⃗=ω⃗×r⃗\vec v=\vec\omega\times\vec r
ω⃗\vec\omega along the axis (right-hand rule); r⃗\vec r from any point on the axis.
Constant angular acceleration
2× asked, last 2021
ω=ω0+αt,θ=ω0t+12αt2,ω2=ω02+2αθ\omega=\omega_0+\alpha t,\qquad \theta=\omega_0t+\tfrac12\alpha t^2,\qquad \omega^2=\omega_0^2+2\alpha\theta
Turns =θ/2π=\theta/2\pi; also n=(f1+f2)t/2n=(f_1+f_2)t/2. Tangential acceleration at=αra_t=\alpha r.
Variable angular acceleration
1× asked, last 2026
ω=dθdt,α=dωdt=ωdωdθ\omega=\frac{d\theta}{dt},\qquad \alpha=\frac{d\omega}{dt}=\omega\frac{d\omega}{d\theta}
Differentiate every term of θ(t)\theta(t) with its sign.
Displacement and average velocity
2× asked, last 2023
∣Δr⃗∣=2Rsin⁡θ2,∣v⃗avg∣v=sin⁡(θ/2)θ/2|\Delta\vec r|=2R\sin\frac\theta2,\qquad \frac{|\vec v_{\text{avg}}|}{v}=\frac{\sin(\theta/2)}{\theta/2}
Constant speed, turned through θ\theta (rad). Distance =Rθ=R\theta.
Change in velocity
2× asked, last 2023
∣Δv⃗∣=v12+v22−2v1v2cos⁡θ∣Δv⃗∣=2vsin⁡θ2 (v1=v2)\begin{aligned}&|\Delta\vec v|=\sqrt{v_1^2+v_2^2-2v_1v_2\cos\theta}\\&|\Delta\vec v|=2v\sin\frac\theta2\ (v_1=v_2)\end{aligned}
θ\theta = angle turned. Quarter turn: v12+v22\sqrt{v_1^2+v_2^2}.
Centripetal acceleration
7× asked, last 2025
ac=v2r=ω2r=ωv,a⃗=−ω2r⃗a_c=\frac{v^2}{r}=\omega^2r=\omega v,\qquad \vec a=-\omega^2\vec r
Towards the centre, perpendicular to v⃗\vec v. Uniform motion: only the speed is constant.

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Practise Circular Motion

28 more formulas in 5 topics

Non-uniform circular motion, Dynamics of uniform circular motion, Banking of roads and conical pendulum, Rotating frame and centrifugal force and Vertical circular motion. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Non-uniform circular motion

  • Tangential and net acceleration2×
  • Equal tangential and centripetal parts2×

Dynamics of uniform circular motion

  • Centripetal force6×
  • Central force F∝r−nF\propto r^{-n}
  • Stone whirled on a string5×
  • Mass on a rotating spring5×
  • Rotating rod or liquid column2×
  • Coin on a turntable3×
  • Car on a level curve4×
  • Over a hump or through a dip

Banking of roads and conical pendulum

  • Banking without friction2×
  • Banking with friction4×
  • Normal force on a banked turn1×
  • Conical pendulum2×
  • Pendulum in a turning vehicle2×
  • Bead on a rotating wire1×

Rotating frame and centrifugal force

  • Centrifugal force5×
  • Ball in a smooth radial groove1×
  • Rotor and death well2×
  • Spinning Earth at the equator1×

Vertical circular motion

  • Tension at any point4×
  • Pendulum released at θ0\theta_0
  • Speeds from energy7×
  • Completing the circle (string)3×
  • Rod or tube
  • Where the string slackens2×
  • Loop from a height1×
  • Leaving a smooth sphere1×

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About 70 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Circular Motion

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