A person on a rotating stool with arms stretched out rotates at an angular speed of 5.00 rad/s. On bringing in his arms, the angular speed increases to 7.50 rad/s. By what factor does the moment of inertia change?

conservation of angular momentum

L₁=L₂
I₁ω₁=I₂ω₂
I₁/I₂=ω₂/ω₁

To find the factor by which the moment of inertia changes, we need to use the concept of conservation of angular momentum.

Angular momentum is given by the product of moment of inertia (I) and angular speed (ω):

L = I * ω

According to the conservation of angular momentum, if no external torques are acting on the person-stool system, the total angular momentum before and after bringing in the arms should be the same.

Initial angular momentum (L1) = Final angular momentum (L2)

Therefore,

I1 * ω1 = I2 * ω2

where I1 and I2 are the initial and final moments of inertia, and ω1 and ω2 are the initial and final angular speeds, respectively.

Given:
ω1 = 5.00 rad/s
ω2 = 7.50 rad/s

Let's assume that the initial moment of inertia is I1, and we want to find the factor by which it changes to I2. We can rewrite the equation as follows:

I1 * ω1 = I2 * ω2

Dividing both sides of the equation by I1 * ω1, we get:

I2 = (I1 * ω1) / ω2

Now we can calculate the factor by which the moment of inertia changes:

Factor = I2 / I1
= ((I1 * ω1) / ω2) / I1
= ω1 / ω2

Plugging in the given values, we have:

Factor = 5.00 rad/s / 7.50 rad/s
= 0.67

Therefore, the moment of inertia changes by a factor of 0.67.

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