Question

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.

Short answer

The average angular velocity is the total angle covered divided by the time taken to cover that angle. Apply the values found in the previous steps into the formula for angular velocity to get the answer.

Step-by-step solution

Calculate Time of Fall

We calculate the time it takes for the diver to fall by using the second equation of motion: \(h=\frac{1}{2}gt^2\), where \(h\) is the height, \(g\) is the gravitational constant 9.8 m/s². So, \(10=\frac{1}{2}*9.8*t^2\) \n Solving this will give us the time of fall.

Calculate the total angle of the rotations

We know the diver completes 2.5 revolutions on the way down. Each revolution is \(2\pi\) radians, so the total angle \(a\) that the diver rotates through is \(2.5 * 2\pi\). This gives us the total angle in radians.

Calculate Angular Velocity

Finally, we apply the formula for average angular velocity, which is the total angle covered divided by the time taken to cover that angle. We can express average angular velocity (\(\omega\)) as \(\omega = \frac{a}{t}\). By substituting the values of \(a\) and \(t\) from previous steps, we get the average angular velocity.

Key concepts

Equations of Motion

The equations of motion are indispensable tools in physics, used to predict the final position and velocity of an object moving under constant acceleration. One such equation, relevant to our problem, is given by the formula for displacement under the influence of gravity:
\[ h = \frac{1}{2}gt^2 \]
In this formula, \( h \) stands for the height traveled, \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and \( t \) is the time taken to travel that height. To find the stop-watch time of a diver's fall, one can rearrange this equation to solve for \( t \), which is crucial for further calculations involved in determining the diver's average angular velocity.

Angular Displacement

Angular displacement refers to the angle through which a point or line has been rotated in a specified sense about a specified axis. It is the angle of the arc traveled at a radius from the axis of rotation, measured in radians or degrees. For a complete revolution, the angular displacement is \( 2\pi \) radians. Since an object can undergo multiple revolutions, the total angular displacement would be multiples of \( 2\pi \).
When the diver completes 2.5 revolutions, we can find the total angular displacement by multiplying the number of revolutions by the angular displacement for one revolution:
\[ a = 2.5 \times 2\pi \]\[ a = 5\pi \text{ radians} \]
This figure is essential for calculating the average angular velocity; it represents the total 'distance' the diver has rotated through.

Rotational Kinematics

Rotational kinematics involves the description of the motion of rotating objects without necessarily addressing forces and mass. It is akin to linear kinematics but for rotation. The primary quantities involved are rotational versions of displacement, velocity, and acceleration. Average angular velocity, which we wish to calculate for the diver, is an analog to linear velocity but in the rotational sense.
Its formula is:
\[ \omega = \frac{a}{t} \]
where \( \omega \) is the average angular velocity, \( a \) is the Angular Displacement, and \( t \) is the time taken to rotate through angle \( a \). To calculate the average angular velocity of our tumbling diver, we use the angular displacement from the 'Angular Displacement' section and the time of fall calculated using the 'Equations of Motion.' By placing these values into the above formula, we obtain the diver's average angular velocity, which tells us the average rate at which he is rotating as he falls toward the water.