Rolling sphere, where to start when i dont know the radius?

[bikkja]

Rolling sphere, problems with the fundementals.

Homework Statement

A sphere with the mass of 2.5 kg rolls without slipping. The speed of the center of gravity is 10 m/s.

a) Calculate the translational kinetic energy of the sphere
b) Calculate the rotational energy of the sphere
c) Calculate the total kinetic energy of the sphere
d) Repeat a)-c) for a hollow cylinder with the same mass and speed

Homework Equations

vT = ω x r ,where vT is the speed of the center of gravity for the sphere

Er = 1/2 x I x ω^2 , where Er is the rotational energy and ω=angular velocity

I = 2/5 x M x r^2 , moment of inertia of the ball

Ek = 1/2 x m x vT^2 + 1/2 x I x ω^2 , where Ek is the total kinetic energy of the rolling motion

The Attempt at a Solution

My first thought was finding the radius using vT=ω x r , my book says that ω=2 rad/s

r = (10 m/s)/(2 rad/s) = 10/2pi m, but I am really not sure if I am allowed to do this. I can't find any examples in my textbook that relates to this problem.

 

[FermiAged]

Rotational energy is usually expressed as a function of ω. Using your Eqn. 2, it should be possible to express as a function of v. Now you can use ratios since the translational energy is known.

Sorry I am somewhat cryptic but I don't want the hall monitors to come down on me.

 

[bikkja]

a)
translational kinetic energy -> Ek = 1/2 x m x vT^2 = 1/2 x 2.5 kg x (10 m/s)^2 = 125 J

b)
rotational energy -> Er = 1/2 x I x ω^2 , I= 2/5m x r^2

Er = 1/2 x 2/5 x m x r^2 x ω^2 , ω=vT / r

Er = 1/5 x m x r^2 x (vT / r)^2

Er = (1/5 x m x r^2 x vT^2)/r^2

Er = 1/5 x m x vT^2 = 1/5 x 2.5 kg x (10 m/s)^2 = 50 J

Am i somewhat close?

 

[tms]

My first thought was finding the radius using vT=ω x r , my book says that ω=2 rad/s.

Is this value of ##\omega## given in the statement of the problem? Is anything else given that you haven't mentioned?

 

[bikkja]

Homework Statement

A sphere with the mass of 2.5 kg rolls without slipping. The speed of the center of gravity is 10 m/s.

a) Calculate the translational kinetic energy of the sphere
b) Calculate the rotational energy of the sphere
c) Calculate the total kinetic energy of the sphere
d) Repeat a)-c) for a hollow cylinder with the same mass and speed

This is the only information that is stated in the problem.

 

[tms]

This is the only information that is stated in the problem.

Where did the value for ##\omega## come from?

 

[bikkja]

I think I misinterpreted an example in my textbook, i haven't use that value in solving the problem above.

 

[tms]

Do you mean you've solved it?

 

[bikkja]

To the best of my current understanding. My textbook have no right/wrong section so i have no way to verify my answer. It does not either have any examples that i find particulary useful helping my solving this problem.

 

[ehild]

a)
translational kinetic energy -> Ek = 1/2 x m x vT^2 = 1/2 x 2.5 kg x (10 m/s)^2 = 125 J

b)
rotational energy -> Er = 1/2 x I x ω^2 , I= 2/5m x r^2

Er = 1/2 x 2/5 x m x r^2 x ω^2 , ω=vT / r

Er = 1/5 x m x r^2 x (vT / r)^2

Er = (1/5 x m x r^2 x vT^2)/r^2

Er = 1/5 x m x vT^2 = 1/5 x 2.5 kg x (10 m/s)^2 = 50 J

Am i somewhat close?

Yes, it is correct. Go ahead.

ehild

 

[bikkja]

c) Total kinetic energy for the sphere:

Ek = Ekt + Er = 125 J + 50 J = 175 J

, is Ekt the correct notation for translational kinetic energy?

 

[ehild]

You can use any notation if you explain what you mean. I would use KE_t and KEr.
And what about question d)?

ehild

 

[bikkja]

Will complete it later today. Thank you so much for your help so far. Great site.

 

[ehild]

Ok, I am looking forward to seeing your solution for d).

ehild