A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minut

NCERT Exemplar · Q14

Q. A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be

(a)

250\pi^2

(b)

100\pi^2

(c)

5\pi^2

(d)

0

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Concept understanding — Kinetic Energy Speed Relation

Kinetic Energy and Speed: The Intuition

Imagine pushing a heavy box across the floor. The harder you push, the faster it moves. But here's the surprising part: doubling the speed does not require double the work — it requires four times the work. That's the core of the kinetic energy–speed relation.

Why? Because kinetic energy isn't about how fast you're moving — it's about how much effort it took to get you moving that fast. And effort (work) depends on both force and distance. When you push something to a higher speed, you have to apply force over a longer distance, and that extra distance multiplies the work required.

Note

Kinetic energy is the energy an object possesses because of its motion. A stationary object has zero kinetic energy.

The Precise Statement

The kinetic energy $K$ of an object of mass $m$ moving with speed $v$ is:

K = \frac{1}{2} m v^2

This is the kinetic energy–speed relation. The key point: kinetic energy is proportional to the square of the speed, not the speed itself.

K = \frac{1}{2} m v^2

Why the Square? A Simple Derivation

Start from Newton's second law: $F = ma$ . If a constant force $F$ acts on an object initially at rest over a displacement $s$ , the work done is $W = F s$ .

From kinematics, for constant acceleration starting from rest: $v^2 = 2 a s$ . So $s = \frac{v^2}{2a}$ .

Substitute into work:

W = F s = (ma) \left( \frac{v^2}{2a} \right) = \frac{1}{2} m v^2

That work becomes the object's kinetic energy. The square comes from the kinematic relation $v^2 = 2as$ — a direct consequence of how distance and speed are linked under constant acceleration.

Watch out

A common mistake: thinking kinetic energy is $\frac{1}{2} m v$ or $m v^2$ . The factor $\frac{1}{2}$ is essential — it comes from the integration of force over distance.

What This Means in Practice

Speed change	Kinetic energy change
Double speed ( $2v$ )	$K$ becomes $4 \times$ original
Triple speed ( $3v$ )	$K$ becomes $9 \times$ original
Halve speed ( $v/2$ )	$K$ becomes $1/4$ of original

This explains why:

A car crash at 100 km/h is four times as destructive as one at 50 km/h (four times the energy to dissipate).
Braking distance quadruples when speed doubles (because brakes must do four times the work).
A bullet at twice the speed penetrates much deeper than twice as far.

Important

Kinetic energy depends only on mass and speed — not on direction. Two objects with the same mass and speed have the same kinetic energy, even if moving in opposite directions.

Units

In SI units:

Mass $m$ in kilograms (kg) …

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A body in circular motion has kinetic energy $\frac{1}{2}mv^2$ where $v = r\omega$ . Converting 300 rpm to angular velocity and using $r = 1\,\text{m}$ , the kinetic energy is $\boxed{250\pi^2\,\text{J}}$ .

Why circular motion still means kinetic energy

When a mass moves in a circle, it is constantly changing direction but its speed remains constant. Kinetic energy depends only on the magnitude of velocity, not its direction. So even though the velocity vector is always turning, the kinetic energy $\frac{1}{2}mv^2$ stays fixed throughout the motion.

The key is to connect the rotational description (revolutions per minute) to the linear speed $v$ that appears in the kinetic energy formula. That connection is $v = r\omega$ , where $\omega$ is the angular velocity in radians per second.

Step-by-step calculation

Convert revolutions per minute to radians per second.

The mass completes 300 revolutions every minute. Each revolution sweeps out $2\pi$ radians, so

\omega = 300 \times 2\pi \,\text{rad/min} = 600\pi \,\text{rad/min}.

Converting to seconds (divide by 60):

\omega = \frac{600\pi}{60} = 10\pi \,\text{rad/s}.

Find the linear speed.

The relationship between linear speed and angular velocity for circular motion is

v = r\omega.

With $r = 1\,\text{m}$ and $\omega = 10\pi\,\text{rad/s}$ : …

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Q.A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be

Concept understanding — Kinetic Energy Speed Relation

Kinetic Energy and Speed: The Intuition

The Precise Statement

Why the Square? A Simple Derivation

What This Means in Practice

Units

Why circular motion still means kinetic energy

Step-by-step calculation

Concept: Kinetic Energy in Circular Motion

Q.A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be

Why circular motion still means kinetic energy

Step-by-step calculation

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