A man stands on a freely rotating platform, as shown, holding an exercise dumbbell in each hand. With outstretched arms, his angular speed is 1.5 rad/s. After he has pulled his arms in, as shown, his angular speed is 5 rad/s. What is the ratio of his final rotational KE to his initial rotational KE? Explain any differences.
Help please. Use the correct equation then substitute the numbers into that equation
moving ars in decreases the moment of inertia, which if momentum is conserved, angular speed increases
momentums:
initialI*1.5=final*5
finalI/initialI=1.5/5
energy:
finalKE/initialKE= .5finalI*5^2/.5InitialI*1.5^2
or = 1.5/5 * (5^2/1.5^2)=5/1.5
What is the answer? Is it 1.998?
To find the ratio of the man's final rotational kinetic energy (KE) to his initial rotational KE, we need to use the equation for rotational KE:
KE = (1/2) I ω^2
Where:
- KE is the rotational kinetic energy
- I is the moment of inertia
- ω is the angular speed (in radians per second)
The moment of inertia depends on the shape and mass distribution of the object. Since we are dealing with a person rotating on a platform and holding dumbbells, we can assume that the moment of inertia remains constant throughout the motion. Thus, we can compare the angular speeds and use the equation directly.
Let's calculate the ratio:
1. When the man has outstretched arms, his initial angular speed (ω_initial) is 1.5 rad/s.
Calculate the initial rotational KE (KE_initial) using the equation:
KE_initial = (1/2) I ω_initial^2
2. When the man pulls his arms in, his final angular speed (ω_final) is 5 rad/s.
Calculate the final rotational KE (KE_final) using the equation:
KE_final = (1/2) I ω_final^2
3. To find the ratio of the final rotational KE to the initial rotational KE, divide the final rotational KE by the initial rotational KE:
Ratio = KE_final / KE_initial
Now let's substitute the given values into the equations and calculate the ratio.
KE_initial = (1/2) I ω_initial^2 = (1/2) I (1.5^2)
KE_final = (1/2) I ω_final^2 = (1/2) I (5^2)
Ratio = KE_final / KE_initial = [(1/2) I (5^2)] / [(1/2) I (1.5^2)]
Now we can simplify the expression by canceling out the common terms:
Ratio = (I / I) * (5^2 / 1.5^2)
Since the moment of inertia is the same for both initial and final angular speeds, I / I equals 1, and the expression simplifies to:
Ratio = (5^2) / (1.5^2)
Further simplification will give you the final result.