Animated Solution for Physics - Kinematics: Spotlight $S$ rotates in a horizontal plane with constant angular velocity of $0.1\text{ rad/s}$. The spot of light $P$ moves along the wall at a distance of $3\text{ m}$. The velocity of the spot $P$ when $\theta =45^{\circ }$ (see fig.) is……. $\text{m/s}$.

The Dance of Light: A Journey into Related Rates

Imagine standing in a dark room, holding a laser pointer. You shine it directly onto a straight wall in front of you. Now, you start rotating your wrist at a steady, constant pace. What happens to the bright spot of light on the wall? It races across the surface. But here is the fascinating part: even though your wrist is turning at a constant speed, the spot on the wall does not move at a constant speed. It actually accelerates! This beautiful interplay between rotational motion and linear motion is a classic physics concept known as 'related rates'. Let's dive into the mathematics behind this phenomenon.

To understand the motion, we first need to freeze time and build a geometric model of our physical reality. Let's define the position of our spotlight as point $S$. We drop a perpendicular line straight from the spotlight to the wall. The problem states that this shortest distance is exactly $3\text{ m}$.

Now, let's look at the moving spot of light on the wall, which we will call $P$. The distance from the foot of our perpendicular line to the spot $P$ is our horizontal position, $x$. Finally, the light beam itself forms the hypotenuse of a right-angled triangle. The angle between the perpendicular line and the light beam is $\theta$.

We now have a perfect right-angled triangle where the adjacent side is $3\text{ m}$, the opposite side is $x$, and the angle is $\theta$.

Our goal is to find the velocity of the spot, which means we need an equation that describes its position $x$. Since we have a right-angled triangle, trigonometry is our best tool. We need a ratio that connects the opposite side ($x$) and the adjacent side ($3$). The tangent function is exactly what we need.

We can write the relationship as:

To make this equation useful for kinematics, we need to isolate the position variable $x$. Multiplying both sides by $3$, we get our master equation for the position of the spot:

Now we transition from static geometry to dynamic kinematics. The spot is moving, meaning its position $x$ is changing with time $t$. The rate of change of position is the linear velocity, $v_P=\frac{dx}{dt}$. Simultaneously, the spotlight is rotating, meaning the angle $\theta$ is changing with time. The rate of change of the angle is the angular velocity, $\omega =\frac{d\theta }{dt}$.

To find the linear velocity, we must differentiate our master equation with respect to time $t$. This is where many students fall into a trap. We are differentiating with respect to time, not $\theta$! Therefore, we must employ the chain rule from calculus.

Differentiating both sides of $x=3\tan \theta$ with respect to $t$ yields:

Applying the chain rule, the derivative of $\tan \theta$ is ${\sec }^2\theta$, multiplied by the derivative of the inner function $\theta$ with respect to $t$:

Now, we substitute our kinematic variables back into the equation. Replacing $\frac{dx}{dt}$ with $v_P$ and $\frac{d\theta }{dt}$ with $\omega$, we get a beautiful formula linking linear and angular velocity:

With our formula ready, we can plug in the specific values given in the problem. We need to find the velocity at the exact instant when the angle $\theta =45^{\circ }$. We are also given that the constant angular velocity $\omega =0.1\text{ rad/s}$.

Substituting these values into our derived formula:

We know from basic trigonometry that $\cos (45^{\circ })=\frac{1}{\sqrt{2}}$. Therefore, its reciprocal, $\sec (45^{\circ })$, is $\sqrt{2}$. Squaring this value gives us ${\sec }^2(45^{\circ })=2$.

Substituting this back into our calculation:

The velocity of the spot of light on the wall at that specific instant is $0.6\text{ m/s}$. This problem beautifully demonstrates how calculus acts as a bridge between static geometry and dynamic motion, allowing us to predict the exact behavior of moving systems.