Q.Velocity vector and acceleration vector of a body in a uniform circular motion are related as
🔒You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Uniform Circular Motion
Uniform Circular Motion
Uniform circular motion is motion along a circular path at constant speed. Although the speed stays the same, the velocity does not — its direction keeps changing at every instant — so the motion has an acceleration even though the speed never changes. This concept covers the kinematics of that motion; the force that causes it (centripetal force) belongs to the Laws of Motion unit.
1. Angular Quantities
As the body sweeps through an angle θ, its angular velocity is
ω=dtdθ
For one full revolution in a period T, at frequency f:
ω=T2π=2πf,T=f1
To convert rpm to rad/s:
ω=602π⋅(rpm)=30π(rpm)
The linear (rim) speed v and the angular speed ω are related by
v=ωr
2. Centripetal Acceleration
Even though the speed is constant, the velocity vector keeps turning — this produces an acceleration directed toward the centre of the circle:
ac=rv2=ω2r=T24π2r=4π2f2r
Use whichever form matches the data you are given. This acceleration is often expressed as a multiple of g (as ac/g, taking g=10 m/s2).
3. Directions
The velocity is always tangential (along the direction of motion); the acceleration is centripetal — radial, pointing inward, and perpendicular to the velocity.
Because the direction of the velocity keeps changing, the change in the velocity vector over an angle Δθ has magnitude
∣Δv∣=2vsin(2Δθ)
So a quarter turn gives ∣Δv∣=v2, a half turn gives 2v, and a full turn gives 0.
The average acceleration over an arc is ∣Δv∣/Δt — this is smaller in magnitude than the instantaneous centripetal acceleration v2/r, and its direction lies along the perpendicular bisector of the chord joining the two points (which, for a circle, always passes through the centre) rather than being radially inward from either endpoint's own position.
4. Points on a Rotating Body
Every point on a rigid rotating body (a wheel, a disc, a clock hand, the Earth) shares the same ω, but the rim speed v=ωr grows with the radius.
- Clock hands: the second hand turns at ω=602π rad/s, the minute hand at 36002π rad/s, the hour hand at 432002π rad/s.
- The Earth spins with ω=864002π≈7.3×10−5 rad/s. A point on the equator moves at ωR≈465 m/s, and a point at latitude λ moves at ωRcosλ (since the circle of latitude has radius Rcosλ).
5. Connected Systems
- Wheels joined by a belt (or two gears in mesh) share the same rim speed, so ω1r1=ω2r2 — the larger wheel turns with the smaller ω.
- Wheels on the same axle (concentric) share the same ω, so the outer rim moves faster (v∝r).
6. Non-Uniform Circular Motion
If the speed also changes, there is a tangential acceleration
at=dtdv=rα
(from the angular acceleration α=dω/dt), in addition to the radial ac=v2/r. These two are perpendicular, so the total acceleration is
a=ac2+at2,tanβ=acat …
In uniform circular motion, the speed stays constant but the direction of velocity keeps changing, which requires a centripetal acceleration directed radially inward toward the centre. The velocity itself, at every instant, is tangential to the circular path.
…
In uniform circular motion the velocity is tangent to the circle while the acceleration points toward the centre, so they are always perpendicular.
In uniform circular motion, the speed |v| is constant but the direction of v keeps changing, which means there must be an acceleration even though the speed doesn't change. This acceleration is the centripetal acceleration, a_c = v^2/r, and it always points toward the centre of the circle (radially inward).
…
Showing the 12 most recent of 18 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.If an object moves with constant speed in a circular path, what can we say about its velocity?(a) It is constant(b) It is changing(c) It is zero(d) It is undefined
›Reveal solutionSolution
Uniform circular motion has constant SPEED but continuously changing VELOCITY, because velocity is a vector that depends on direction too.
Velocity has both magnitude (speed) and direction. In circular motion at constant speed, the object's direction of travel is always tangential to the circle, and this tangent direction keeps rotating as the object goes around. So even though |v| stays the same, the vector v (pointing along the tangent) is continuously changing.
…
- CBSE 2026Set ANNUAL1 markMCQQ.What is the magnitude of centripetal acceleration for an object moving in a circle of radius r with speed v?(a) a = v/r(b) a = r/v(c) a = v/r^2(d) a = r^2/v
›Reveal solutionSolution
The true formula for centripetal (radial) acceleration in circular motion is a = v^2 / r. None of the four printed options exactly reproduces this; option (a) is chosen as the intended answer since it is structurally closest (v in the numerator, r in the denominator) and most consistent with a lost superscript.
For an object moving in a circle of radius r with constant speed v, the centripetal acceleration (directed toward the centre) is derived from the rate of change of the velocity vector's direction:
a_c = v^2 / r
…
- CBSE 2026Set ANNUAL1 markMCQQ.If a particle executes uniform circular motion, then _________.(a) its velocity and the acceleration are constant(b) its velocity and the speed are constant(c) its speed and the magnitude of acceleration are constant(d) its acceleration and the speed are constant
›Reveal solutionSolution
Uniform circular motion has constant speed and constant magnitude of centripetal acceleration, but both the velocity and the acceleration continuously change direction, so they are not themselves constant vectors.
In uniform circular motion, a particle moves on a circular path with a constant speed. Let's examine each quantity:
-
Velocity: This is a vector with both magnitude and direction. Although the magnitude (speed) stays fixed, the direction of velocity is always tangential to the circle and keeps changing as the particle moves around. So velocity is NOT constant.
-
Speed: This is just the magnitude of velocity, and by definition of 'uniform' circular motion, this stays constant throughout the motion.
…
-
- CBSE 2025Set ANNUAL1 markMCQQ.What is the magnitude of centripetal acceleration for an object moving in a circle of radius r with speed v?(a) a = v/r(b) a = r/v(c) a = v^2/r(d) a = r^2/v
›Reveal solutionSolution
Centripetal acceleration, which keeps an object moving in a circle by constantly redirecting its velocity toward the centre, is given by a = v^2/r.
For an object moving in a circle of radius r with constant speed v, the velocity direction continuously changes even though the speed doesn't. This change in direction produces an acceleration directed toward the centre of the circle, called centripetal …
- CBSE 2025Set ANNUAL1 markMCQQ.Match the following - Column A item: Centripetal acceleration. Pick the matching relation from Column B.(a) F.V (Force . Velocity)(b) T is proportional to sqrt(l)(c) eta is proportional to 1/(dv/dx)(d) mu_s = tan(theta)(e) Y is proportional to 1/l(f) v^2/r
›Reveal solutionSolution
Centripetal acceleration is given by a_c = v^2/r.
For a particle moving with uniform speed v along a circular path of radius r, its velocity direction constantly changes even though its speed does not. This produces an acceleration directed towards the centre of the circle, called centripetal acceleration, with magni …
- CBSE 2024Set ANNUAL1 markMCQQ.A particle is moving with uniform speed of V on a circular path of radius r. What will be the average acceleration in the interval of time in which particle describes half the circle? (A) V^2/r (B) 2V^2/r (C) 2V^2/(πr) (D) V^2/(πr)
›Reveal solutionSolution
Average acceleration over half a circular revolution is 2V2/(πr).
At the start of the half-circle the velocity is v1=V in some direction; after half a revolution the particle moves diametrically opposite, so v2=−V along the original direction. The change in velocity is ∣Δv∣=∣−V−V∣=2V.
…
- CBSE 2024Set ANNUAL1 markMCQQ.If an object moves with constant speed in a circular path, what can we say about its velocity?(a) It is constant(b) It is changing(c) It is zero(d) It is undefined
›Reveal solutionSolution
Velocity = speed + direction. In uniform circular motion, speed is constant but direction changes every instant, so velocity is always changing.
A particle in uniform circular motion has its velocity vector always tangent to the circle. As the particle moves around the circle, the direction of this tangent keeps changing, even though its magnitude (the speed) stays fixed. Since velocity is a vector quantity, a change in direction alone means the v …
- CBSE 2024Set SET-AP55001 markMCQQ.In uniform circular motion:(a) Both velocity and acceleration change(b) Both velocity and acceleration are constant(c) Velocity remains constant and acceleration changes(d) Acceleration remains constant and velocity changes
›Reveal solutionSolution
In uniform circular motion, speed and the magnitude of acceleration stay constant, but velocity and acceleration are VECTORS whose direction changes every instant — so both are said to 'change'.
Consider a particle moving on a circle of radius r with constant speed v. At every point, its velocity is tangent to the circle, and as the particle moves around, the direction of this tangent keeps rotating. So even though |v| is fixed, the velocity vector v (which has both magnitude and direction) is continuously changing.
…
- CBSE 2023Set ANNUAL1 markMCQQ.Velocity vector and acceleration vector of a body in a uniform circular motion are related as(1) both in the same direction(2) perpendicular to each other(3) both in opposite directions(4) not related to each other
›Reveal solutionSolution
In uniform circular motion the velocity is tangent to the circle while the acceleration points toward the centre, so they are always perpendicular.
In uniform circular motion, the speed |v| is constant but the direction of v keeps changing, which means there must be an acceleration even though the speed doesn't change. This acceleration is the centripetal acceleration, a_c = v^2/r, and it always points toward the centre of the circle (radially inward).
…
- CBSE 2023Set ANNUAL1 markQ.Answer in one word/sentence: Write the formula for centripetal force for circular motion.
›Reveal solutionSolution
Centripetal force, the force that keeps a body moving in a circle, is given by F = mv^2/r (equivalently F = momega^2*r).
For an object of mass m moving with constant speed v along a circular path of radius r, its velocity direction is continuously changing, which means it has a centripetal (centre-seeking) acceleration a = v^2/r, directed toward the centre of the circle. By Newton's second law, the net force …
- CBSE 2023Set ANNUAL1 markMCQQ.Force acting uniformly on an object doing circular motion is:(a) centripetal force(b) atomic force(c) internal force(d) gravitational force
›Reveal solutionSolution
The force in uniform circular motion is the centripetal force.
A body in uniform circular motion has constant speed but continually changing direction, so it is accelerating toward the centre (centripetal acceleration v^2/r). By Newton's second law a …
- CBSE 2022Set TERM11 markMCQQ.A body is travelling in a circle at a constant speed. It(1) has an inward acceleration(2) has constant velocity(3) has no acceleration(4) has an outward radial acceleration
›Reveal solutionSolution
Constant speed on a circular path does not mean zero acceleration -- the direction of velocity keeps changing, and this change requires a centripetal (inward) acceleration, v^2/r, at every instant.
Even though the SPEED (magnitude of velocity) is constant, the VELOCITY (a vector) is continuously changing direction as the body goes around the circle. Since acceleration is the rate of change of the velocity VECTOR, a change in direction alone is enough to produce a nonzero acceleration.
…
🎓Unlock everything free for 14 days
- ✓Full step-by-step solutions
- ✓Concept-first explanations
- ✓Methods, shortcuts & mistakes
- ✓PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.