A diver can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5s when in the tuck position, what is the percent change in her kinetic energy as she changes to the straight position??
Please explain to me how to get this problem step by step.
The kinetic energy can increase because of muscular work done going into the tuck position.
Remember that I * w remains constant. Angular momentum conservation is not violated.
If the momen t of inertia I decreases by a factor of 3.5,then the angular frequency w increases by a factor 3.5, and the kinetic energy, (1/2)I w^2, increases by a factor
(1/3.5)*(3.5)^2 = 3.5
The percent change in KE is therefore +250%
How is it +250%?
To solve this problem, we need to follow these steps:
Step 1: Find the change in moment of inertia from the tuck position to the straight position.
Step 2: Calculate the change in kinetic energy using the formula for rotational kinetic energy.
Step 3: Determine the percent change in kinetic energy.
Let's now go through each step in detail:
Step 1: Find the change in moment of inertia
We are given that the diver can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. This means the moment of inertia in the straight position is 3.5 times greater than in the tuck position.
Let I_tuck be the moment of inertia in the tuck position.
Let I_straight be the moment of inertia in the straight position.
We can write the equation as:
I_straight = 3.5 * I_tuck
Step 2: Calculate the change in kinetic energy
The formula for rotational kinetic energy is given by:
K_rotational = (1/2) * I * ω^2
Let K_tuck be the kinetic energy in the tuck position.
Let K_straight be the kinetic energy in the straight position.
We can write the equation as:
K_straight - K_tuck = (1/2) * (I_straight * ω_straight^2 - I_tuck * ω_tuck^2)
Since we are not given the values of ω_straight and ω_tuck, we need additional information to calculate the difference in kinetic energy.
Step 3: Determine the percent change in kinetic energy
To find the percent change in kinetic energy, we can use the following formula:
Percent change = ((final - initial) / initial) * 100
Using the equation from Step 2, the percent change in kinetic energy would be:
Percent change = ((K_straight - K_tuck) / K_tuck) * 100
To calculate this percentage, we will need the values of K_straight and K_tuck from either given data or additional information.
So, without additional information, we can only go up to Step 2 and determine the expression for the change in kinetic energy.
To solve this problem, we'll need to use the concepts of moment of inertia and kinetic energy.
Step 1: Understand the problem
The problem provides two key pieces of information:
- A diver reduces her moment of inertia by a factor of 3.5 when changing from the straight position to the tuck position.
- In the tuck position, the diver makes 2.0 rotations in 1.5 seconds.
We need to find the percent change in kinetic energy as the diver changes from the tuck position to the straight position.
Step 2: Calculate the moment of inertia
The moment of inertia is a measure of an object's resistance to changes in its rotation. The moment of inertia for two positions (tuck and straight) can be represented as I_tuck and I_straight.
Given that the moment of inertia is reduced by a factor of 3.5 when changing from the straight position to the tuck position, we can write: I_straight = 3.5 * I_tuck.
Step 3: Calculate the angular velocity
In the tuck position, the diver makes 2.0 rotations in 1.5 seconds. The angular velocity (ω) can be calculated using the formula:
ω = (2π * rotations) / time
Substituting the values, we have: ω_tuck = (2π * 2.0) / 1.5.
Step 4: Calculate kinetic energy
The kinetic energy of a rotating object can be calculated using the formula:
Kinetic energy (KE) = (1/2) * moment of inertia * angular velocity^2
In the tuck position, the kinetic energy (KE_tuck) can be calculated using the moment of inertia and angular velocity in that position. Similarly, the kinetic energy (KE_straight) in the straight position can be calculated using the new moment of inertia and angular velocity.
Step 5: Calculate the percent change in kinetic energy
The percent change in kinetic energy can be calculated using the formula:
Percent change = ((new value - old value) / old value) * 100
In this case, the old value is KE_tuck, and the new value is KE_straight.
Now that we have calculated the kinetic energies in both positions, we can use the given formula to find the percent change.
I hope this helps!