Finding angular speed given T and I

[lespiderboris]

Homework Statement

A constant torque of 25.5 N · m is applied to a grindstone whose moment of inertia is
0.117 kg · m2.
Using energy principles, and neglecting friction, find the angular speed after the grind-
stone has made 14.6 rev.
Answer in units of rad/s

Homework Equations

1 rad = 1/2π rev (π = pi)

T=Iα (α = alpha = angular acceleration)(T=torque)

α=Δω/t

The Attempt at a Solution

All I've got it that 14.6 rev = 91.7345...rad.
... I think those are the only relevant equations since we don't know the radius or the mass OR the initial velocity of the grindstone. (Do we assume it was at rest? Does it even matter?) I must be wrong though as I am stumped on how to solve this. Thank you to anyone able and kind enough to help!

 

[nasu]

The text indicate that you are expected to use "energy principles".
Do you know the work-energy theorem?
And yes, you can assume it starts from rest.

 

[lespiderboris]

I know I₁ω₁=I₂ω₂

And that the total angular momentum of a system is conserved, i.e. remains constant.

Is that where you're leading me? :)

 

[nasu]

These are angular momenta and not energies.
And you have torque so the angular momentum is not conserved.
No, the work-energy theorem states that the change in kinetic energy is equal to the work done by the external force. In this case, taking the initial KE =0, you will have final KE=work.
You will have to express this in angular quantities and solve to find the angular velocity.

 

[Nickie]

I would approach this problem by first understanding the given information and determining what is needed to solve for the angular speed. The given information includes a constant torque of 25.5 N · m and the moment of inertia of the grindstone, which is 0.117 kg · m2. We are also told to neglect friction and are given the number of revolutions made by the grindstone, which is 14.6.

To solve for the angular speed, we can use the equation T=Iα, where T is the torque, I is the moment of inertia, and α is the angular acceleration. Since we are neglecting friction, we can assume that the torque is the only force acting on the grindstone, and therefore, it is equal to the net torque. We can also assume that the grindstone is initially at rest, so its initial angular velocity is 0.

Using the equation α=Δω/t, where Δω is the change in angular velocity and t is the time, we can rewrite the equation T=Iα as T=I(Δω/t). We know that the grindstone has made 14.6 revolutions, which is equivalent to 91.7345... radians, so we can plug this value in for Δω. We also know that the time is the same for every revolution, so we can use the time for one revolution, which is equal to the period (T) of the grindstone.

Therefore, the equation becomes 25.5 = 0.117(91.7345...)/T. Solving for T, we get T = 0.0485 seconds. Now, we can use the equation ω=Δθ/Δt, where ω is the angular velocity, Δθ is the change in angular displacement, and Δt is the change in time. Since we know that the grindstone has made 14.6 revolutions, we can plug in 91.7345... radians for Δθ and 0.0485 seconds for Δt. Solving for ω, we get an angular speed of approximately 189.4 rad/s.

In conclusion, using energy principles and neglecting friction, we were able to find the angular speed of the grindstone after it has made 14.6 revolutions. This problem demonstrates the application of equations in rotational motion and the importance of understanding the given information in order to solve