If Minute hand of a clock is 1 cm long,what will be the change in velocity of its tip after 15 seconds ??

To determine the change in velocity of the tip of a minute hand after 15 seconds, we first need to understand how the minute hand moves and how to calculate its velocity. The minute hand completes one full revolution around the clock face in 60 minutes, which is equivalent to 360 degrees. Let's break this down step by step.

Understanding Circular Motion

The minute hand moves in a circular path, and its tip traces a circle with a radius equal to the length of the minute hand. In this case, the radius is 1 cm. The circumference of the circle, which represents the distance traveled by the tip of the minute hand in one complete revolution, can be calculated using the formula:

• Circumference = 2 × π × radius

Substituting the radius:

• Circumference = 2 × π × 1 cm ≈ 6.28 cm

Calculating the Angular Velocity

The minute hand completes one full revolution (360 degrees) in 60 minutes, or 3600 seconds. The angular velocity (ω) can be calculated as:

• ω = Total angle / Time = 360 degrees / 3600 seconds = 0.1 degrees/second

To convert this into radians (since most calculations in physics use radians), we use the conversion factor (π radians = 180 degrees):

• ω = 0.1 degrees/second × (π radians / 180 degrees) ≈ 0.00175 radians/second

Finding Linear Velocity

The linear velocity (v) of the tip of the minute hand can be calculated using the formula:

• v = r × ω

Substituting the values we have:

• v = 1 cm × 0.00175 radians/second ≈ 0.00175 cm/second

Change in Velocity After 15 Seconds

Now, we need to find the change in velocity after 15 seconds. In circular motion, the speed (magnitude of velocity) remains constant, but the direction of the velocity changes continuously. After 15 seconds, the minute hand will have moved:

• Angle moved = ω × time = 0.1 degrees/second × 15 seconds = 1.5 degrees

To find the new position of the tip of the minute hand, we can visualize the change in direction. The initial velocity vector points in one direction, and after 15 seconds, it will point 1.5 degrees from that initial position. The magnitude of the velocity remains the same, but the direction has changed.

Calculating the Change in Velocity Vector

To find the change in velocity, we can use vector subtraction. The initial velocity vector can be represented as:

• V_initial = (0.00175 cm, 0) [pointing along the positive x-axis]

After 15 seconds, the new velocity vector can be represented as:

• V_final = (0.00175 cm × cos(1.5°), 0.00175 cm × sin(1.5°))

Calculating these components:

• V_final ≈ (0.00175 cm × 0.998, 0.00175 cm × 0.0262) ≈ (0.00174 cm, 0.000046 cm)

Now, to find the change in velocity (Δv), we subtract the initial velocity vector from the final velocity vector:

• Δv = V_final - V_initial

Calculating this gives us:

• Δv ≈ (0.00174 cm - 0.00175 cm, 0.000046 cm - 0) ≈ (-0.00001 cm, 0.000046 cm)

Magnitude of Change in Velocity

The magnitude of the change in velocity can be found using the Pythagorean theorem:

• |Δv| = √((-0.00001 cm)² + (0.000046 cm)²) ≈ 0.000048 cm

In summary, the change in velocity of the tip of the minute hand after 15 seconds is approximately 0.000048 cm. This illustrates how even though the speed remains constant, the continuous change in direction results in a change in velocity, which is a fundamental concept in circular motion.