A diver makes 2.5 complete revolutions on the way from a 10-m platform to the water below. Assuming zero initial vertical velocity, calculate the average angular velocity for this drive

To find the average angular velocity of the diver, we can use the relationship between angular displacement and time. In this case, the diver makes 2.5 complete revolutions while descending from a height of 10 meters. Let’s break down the steps to calculate the average angular velocity.

Understanding Angular Velocity

Angular velocity measures how quickly an object rotates about an axis. It is defined as the change in angular displacement over time. The formula to calculate average angular velocity (ω) is:

ω = Δθ / Δt

Where:

• Δθ = change in angular displacement (in radians)
• Δt = change in time (in seconds)

Step 1: Calculate Angular Displacement

Since the diver makes 2.5 complete revolutions, we first need to convert revolutions into radians. One complete revolution is equal to 2π radians, so:

Δθ = 2.5 revolutions × 2π radians/revolution = 5π radians

Step 2: Determine the Time Taken

Next, we need to find out how long it takes for the diver to reach the water. To do this, we can use the physics of free fall. The formula for the distance fallen under gravity is:

d = 0.5 * g * t²

Where:

• d = distance (10 meters)
• g = acceleration due to gravity (approximately 9.81 m/s²)
• t = time in seconds

Rearranging the formula to solve for time, we get:

t² = 2d / g

Plugging in the values:

t² = 2 * 10 m / 9.81 m/s² ≈ 2.04

Now, take the square root to find t:

t ≈ √2.04 ≈ 1.43 seconds

Step 3: Calculate Average Angular Velocity

Now that we have both the angular displacement and the time, we can calculate the average angular velocity:

ω = Δθ / Δt = 5π radians / 1.43 seconds

Calculating this gives:

ω ≈ 10.99 radians/second

Final Thoughts

The average angular velocity of the diver as they perform 2.5 revolutions while descending from a 10-meter platform is approximately 10.99 radians per second. Understanding how to relate angular motion to linear motion is crucial in physics, especially in fields involving rotational dynamics.