We can roughly model a gumnastic tumbler as a uniform solid cylinder of mass 75 kg and diameter 1.0 m. If the tumbler rolls forward at .50 rev/s. A) how much total kinetic energy does he have and B) what percent of his total kinetic enery is rotational?

I for the cylinder is I = 1/2MR^2

I still don't understand how to get the total KE through TranKE=1/2 m v^2

because 1/2 (75 kg)(.50 rev/s) = 9.375 and the answer is coming up in J in the back of the book

Total KE=translationalKE + rotationalKE

= 1/2 m v^2 + 1/2 I w^2

To calculate the total kinetic energy and the percentage of rotational kinetic energy for the gymnastic tumbler, we need to consider both its translational and rotational motion.

A) Total Kinetic Energy:
The total kinetic energy (KE) is the sum of the translational kinetic energy (KE_trans) and the rotational kinetic energy (KE_rot).

1. Translational Kinetic Energy (KE_trans):
The translational kinetic energy of a rolling object can be determined using the formula:

KE_trans = (1/2) * m * v^2

Where:
m = mass of the tumbler
v = linear velocity of the tumbler (which can be calculated using the formula v = ω * r)
ω = angular velocity of the tumbler (given as 0.50 rev/s)
r = radius of the tumbler (which is half of its diameter)

Since the diameter of the tumbler is given as 1.0 m, the radius would be 0.5 m.

v = ω * r = 0.50 rev/s * 2π rad/rev * 0.5 m = π m/s

Plugging in the values, we have:

KE_trans = (1/2) * 75 kg * (π m/s)^2

Calculate the value to find the translational kinetic energy.

2. Rotational Kinetic Energy (KE_rot):
The rotational kinetic energy of a rolling object can be calculated using the formula:

KE_rot = (1/2) * I * ω^2

Where:
I = moment of inertia of the tumbler (for a solid cylinder, I = (1/2) * m * r^2)
ω = angular velocity of the tumbler

Plugging in the values, we have:

I = (1/2) * 75 kg * (0.5 m)^2

Calculate the value of I.

KE_rot = (1/2) * I * (ω)^2

Calculate the value of KE_rot using the moment of inertia for the given solid cylinder.

The total kinetic energy would be the sum of KE_trans and KE_rot.

B) Percentage of Rotational Kinetic Energy:
To find the percentage of rotational kinetic energy, you can divide the rotational kinetic energy (KE_rot) by the total kinetic energy and multiply by 100.

Rotational percentage = (KE_rot / (KE_trans + KE_rot)) * 100

Plug in the values to calculate the percentage of rotational kinetic energy.

By following these steps and performing the necessary calculations, you can find the answers to parts A and B of the question.

I = 1/2MR^2

if the angular rotation is 1/2 rev/sec, then the forward speed is 1/2*2PI*.5 m/s

Look up the moment of inertia I for a cylinder.

TranKE=1/2 m v^2
RotationalKE=1/2 I w^2 where w is PI r/sec