Building a study guide--please show all work so I can follow easily. Thank you!

Question

Building a study guide--please show all work so I can follow easily. Thank you!

Suppose a bicycle tire (including its hub and spokes) has a moment of inertia given by It = (4/5)MR^2 , where M is the mass of the tire and R is the radius of the tire. The tire rolls without slipping down a ramp of length 20.0 m which makes an angle of 10.0 degrees with the horizontal. You can neglect air resistance in this problem.

(b) What is the total kinetic energy of the tire (translational plus rotational) when it reaches the bottom of the ramp, and how much rotational work is done during the motion of the tire when it reaches this point?

Explanation / Answer

So first we should determine the total energy at the top of ramp where it is at rest. There will be only potential energy due to gravity.
So first we need to find the height. We know the ramp is 20m long at an angle of 10 degrees so we can use simple geometry.
h=20sin(10)
h=3.4732 m
Now lets find the total energy
TE=PE=mgh=M(9.8)(3.4732
PE=34.035M
This will also equal the total kinetic energy

KE=34.035M
Now we need to relate this to the kintec energy to both rotation and translational the equation will be as follow:
KE=(1/2)I2+(1/2)mv2

We can substitue v/r for :

KE=(1/2)I(v/r)2+(1/2)mv2

Now plug in what we know

34.035M=(2/5)MR2v2/R2+(1/2)Mv2

34.025=(9/10)v2

v=6.14953 m/s

The rotational work will just be equal to the rotational kinetic energy which we already reduced a little so we can extract it from 3 equations above

W=(2/5)Mv2

W=15.13M joules

Hope that helps

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