Question

A constant torque of 25.0 N⋅m is applied to a grindstone whose moment of inertia is 0.130 kg⋅m². Using energy principles and neglecting friction, find the angular speed after the grindstone has made 15.0 revolutions. Hint: The angular equivalent of Wnet=FΔx=1/2 Mv²−1/2 Mv²ᵢ is Wnet=Δs=1/2Iw² −1/2Iw²ᵢ

Answer 1

The angular speed of the grindstone after it has made 15.0 revolutions is 11.18 rad/s.

Step-by-step explanation:

To find the angular speed of the grindstone after it has made 15.0 revolutions, we can use the equation for work and energy.

From the information given, the net work done on the grindstone can be calculated using the equation: net work =
`ΔKE_rot = 1/2 * I * (ω^2 - ω_i^2)`

Where I is the moment of inertia of the grindstone, ΔKE_rot is the change in rotational kinetic energy, ω is the final angular speed, and ω_i is the initial angular speed.

Substituting the given values, we have:
`25.0 N·m = 1/2 * 0.130 kg·m^2 * (ω^2 - 0)`

`ω^2 = 2 * (25.0 N·m) / (0.130 kg·m^2)`

`ω = √[(2 * (25.0 N·m) / (0.130 kg·m^2))]`

`ω = 11.18 rad/s`

Answer 2

To find the angular speed after the grindstone has made 15.0 revolutions, we can use energy principles and the equation net W = (1/2)Iω² - (1/2)Iω₀². By calculating the work done on the grindstone and equating it to the change in rotational kinetic energy, we can solve for the final angular speed and the final rotational kinetic energy.

Step-by-step explanation:

To find the angular speed after the grindstone has made 15.0 revolutions, we can use energy principles. The work done on the grindstone can be calculated using the equation net W = (1/2)Iω² - (1/2)Iω₀², where net W is the work done, I is the moment of inertia of the grindstone, ω is the final angular speed, and ω₀ is the initial angular speed. We know the torque applied to the grindstone is 25.0 N⋅m and the moment of inertia is 0.130 kg⋅m². Additionally, 15.0 revolutions can be converted to radians by multiplying by 2π, giving us the rotation angle.

Using the given information, we can calculate the work done on the grindstone. Then, by equating the work done to the change in rotational kinetic energy, we can solve for the final angular speed. The final rotational kinetic energy can also be calculated using the same equation.