Q.An insect trapped in a circular groove of radius $12\ \text{cm}$ moves along the groove steadily and completes $7$ revolutions in $100\ \text{s}$.

Concept: Angular Velocity — the rate of change of angular displacement, related to linear speed by $v = \omega r$.

(a) The insect completes $7$ revolutions in $100\ \text{s}$. Each revolution is $2\pi$ radians, so the total angular displacement is $7 \times 2\pi = 14\pi\ \text{rad}$.

Angular speed:

Linear speed:

$v = \omega r = (0.14\pi)(12) = 1.68\pi\ \text{cm/s}$ …

Why angular velocity is the natural starting point

When an object moves along a circular path at a steady rate, the most direct way to describe its motion is through angular velocity — the rate at which the angular position changes. Unlike linear velocity, which depends on the path's radius, angular velocity is a property of the rotation itself. Once you know how many revolutions happen per second, you have the angular speed. Linear speed then follows immediately from $v = \omega r$.

The acceleration in uniform circular motion is purely centripetal — always pointing toward the centre. Its magnitude is constant, but its direction changes every instant. That distinction (constant magnitude vs. constant vector) is the key to part (b).

1. Find the angular speed

The insect completes 7 revolutions in 100 seconds. One revolution corresponds to an angle of $2\pi$ radians. So the total angle covered is:

Angular speed is angle per unit time:

Evaluating numerically:

2. Compute the linear speed

Linear speed $v$ is related to angular speed by $v = \omega r$, where $r = 12\ \text{cm}$.

3. Is the acceleration vector constant?

In uniform circular motion, the acceleration is centripetal — always directed toward the centre of the circle. As the insect moves, the direction from the insect to the centre changes continuously. Therefore, the acceleration vector changes direction at every point along the path. …

Concept: Centripetal Acceleration from Differentiating a Rotating Position Vector

Method: Vector-Calculus Derivation ($\vec a = -\omega^2\vec r$), Not a Qualitative Direction Argument

Rather than arguing in words that "the direction to the centre keeps changing, so $\vec a$ is not constant," this method writes the insect's position as an explicit rotating vector, differentiates it twice using calculus, and reads the non-constancy of $\vec a$ directly off the resulting formula -- which also reproduces the magnitude $a=\omega^2r$ as a byproduct.

Step 1 -- Angular speed (same first step, unavoidable)

7 revolutions in $100\ \text{s}$ means an angular displacement of $7\times2\pi = 14\pi$ rad:

Step 2 -- Write the position vector as a function of time

Place the centre of the groove at the origin. With radius $r=12\ \text{cm}$ and angular speed $\omega$, the insect's position at time $t$ is:

Step 3 -- Differentiate once to get velocity

(Confirms the linear-speed relation $v=\omega r$ directly from calculus, not just by quoting the formula.)

Step 4 -- Differentiate again to get acceleration

This bracket is exactly $\vec r(t)$ from Step 2 -- so:

Step 5 -- Read the answer to "is $\vec a$ constant?" directly off this formula …