Problem 11 - Master Resource Book in JEE Main Physics

Question

The length of second's hand in a watch is \(1 \mathrm{~cm}\). The change in velocity of its tip in \(15 \mathrm{~s}\) is (a) zero (b) \(\frac{\pi}{30 \sqrt{2}} \mathrm{~cm} / \mathrm{s}\) (c) \(\frac{\pi}{30} \mathrm{~cm} / \mathrm{s}\) (d) \(\frac{\pi \sqrt{2}}{30} \mathrm{~cm} / \mathrm{s}\)

Step-by-Step Solution

Answer

The change in velocity is \(\frac{\pi\sqrt{2}}{30}\, \mathrm{cm/s}\), which is option (d).

Step 1: Understand the Problem

We are asked to find the change in velocity of the tip of a second's hand in a watch over 15 seconds. The length of the hand is given as 1 cm.

Step 2: Determine Initial and Final Velocities

The second's hand completes one full rotation in 60 seconds, which means it moves with uniform circular motion. Initially, after 15 seconds, the tip will have moved from the 12 o'clock position to the 3 o'clock position. The velocities at these positions are perpendicular to the radial line to the center.

Step 3: Calculate the Angular Velocity

The angular velocity \( \omega \) for the second's hand is given by \( \omega = \frac{2\pi}{60} \) radians per second, as it completes one full rotation (\(2\pi\) radians) in 60 seconds.

Step 4: Compute Initial and Final Velocities

The magnitude of the linear velocity \( v \) is given by \( v = \omega r \). Since \( r = 1 \) cm and \( \omega = \frac{\pi}{30} \), the velocity is \( v = \frac{\pi}{30} \). Both initial and final velocities have this magnitude but different directions.

Step 5: Find the Change in Velocity

The change in velocity \( \Delta v \) is the vector difference between the final \( \vec{v}_f \) and initial \( \vec{v}_i \) velocities. Because they are perpendicular (one moves eastward, the other moves northward), use Pythagorean theorem: \( \Delta v = \sqrt{v^2 + v^2} = v\sqrt{2} \).

Step 6: Calculate \( \Delta v \)

Substitute \( v = \frac{\pi}{30} \) into the expression from Step 5: \( \Delta v = \frac{\pi}{30} \sqrt{2} \).

Step 7: Match with the Given Options

Look at the given options and match \( \Delta v = \frac{\pi\sqrt{2}}{30} \) with the correct choice. This corresponds to option (d).

Key Concepts

Angular Velocity

Angular velocity is a concept that comes into play when we're dealing with circular motion. It describes how fast an object rotates or spins around a fixed point. Specifically, it is the rate of change of the angular position of an object with respect to time. Think of it as the speed at which an object is moving along a circular path.

In mathematical terms, angular velocity is represented by the Greek letter omega (\( \omega \)). For a circular path, like the second's hand of a watch, angular velocity can be calculated using the formula:

• \( \omega = \frac{\text{Angle} \; \text{in} \; \text{radians}}{\text{Time} \; \text{in} \; \text{seconds}} \)

The second's hand of a watch completes one full circle, or \( 2\pi \) radians, in 60 seconds. Hence, the angular velocity of the second's hand is \( \frac{2\pi}{60} \) radians per second, which simplifies to \( \frac{\pi}{30} \) radians per second. This uniform rate makes it straightforward to understand how sections of time, like 15 seconds, relate to its movement around the clock face.

Linear Velocity

Linear velocity is closely related to angular velocity, but instead of thinking about rotation, we think about actual speed along a linear path. It's a measure of how quickly something is moving through space.

For any object moving in a circle, we can find its linear velocity by multiplying its angular velocity by the radius of the circle. The formula is:

• \( v = \omega \cdot r \)

In the case of the second's hand of the watch, the radius \( r \) is the length of the hand, which is 1 cm. Using our previously calculated angular velocity of \( \frac{\pi}{30} \), the linear velocity \( v \) becomes \( \frac{\pi}{30} \times 1 = \frac{\pi}{30} \) cm/s.

Linear velocity helps us understand the actual speed of the tip of the second's hand as it moves around its circular path, making it a critical part of determining changes in velocity as seen in this exercise.

Vector Addition

When dealing with motions in different directions, like those of the second's hand, we use vector addition to find resultant velocities or changes. Vectors have both magnitude and direction, making them perfect for representing velocity towards different points in the circular path.

In this problem, after 15 seconds, the initial and final velocities of the second's hand tip are perpendicular to each other, because the hand moves from the 12 o'clock to the 3 o'clock position. Using vector addition, specifically the Pythagorean theorem, allows us to find the change in velocity (\( \Delta v \)).

The formula is:

• \( \Delta v = \sqrt{v^2 + v^2} = v\sqrt{2} \)

Here, each velocity \( v \) is \( \frac{\pi}{30} \). Substituting into the formula gives \( \Delta v = \frac{\pi\sqrt{2}}{30} \) cm/s, demonstrating how vector addition provides the solution to the change in velocity for this exercise.