Help Me Solve Rotational Kinetic Energy Problem

[brad sue]

Hi Please I would like to have some help with this problem:
A cylinder of mass .2 kg rolls without slipping down an inclined plane of 15 degree.
What is its rotational kinetic energy after it rolls 80cm?
I tried the formula W=K_f - K_o but It does not bring me to the solution.
Thank you for our help

 

[Tide]

Use energy conservation and note that the cylinder has both translational and rotational kinetic energy. You'll need to use a little geometry to determine the change in elevation of the cylinder and don't forget to use the moment of inertia in your rotational KE calculation.

 

[quicknote]

Sure, I'd be happy to help you with this problem. First, let's start by breaking down the problem into smaller steps.

Step 1: Determine the initial and final kinetic energies
The formula you mentioned, W = Kf - Ko, is correct. W represents the work done on the cylinder, Kf represents the final kinetic energy, and Ko represents the initial kinetic energy. We know that the cylinder starts with zero velocity and then rolls down the inclined plane, so the initial kinetic energy is zero. To find the final kinetic energy, we need to use the formula for kinetic energy, K = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity.

Step 2: Find the moment of inertia
The moment of inertia for a cylinder rolling without slipping is I = 1/2 * m * r^2, where m is the mass of the cylinder and r is the radius. In this problem, m = 0.2 kg and r = 80 cm = 0.8 m. Plugging these values into the formula, we get I = 1/2 * 0.2 kg * (0.8 m)^2 = 0.064 kg*m^2.

Step 3: Calculate the final angular velocity
To find the final angular velocity, we need to use the formula for rotational motion, ω = v/r, where v is the linear velocity and r is the radius. In this problem, v is the velocity of the cylinder as it rolls down the inclined plane, which we can find using the formula for linear motion, v = √(2gh), where g is the acceleration due to gravity (9.8 m/s^2) and h is the height of the inclined plane. In this problem, h = 80 cm = 0.8 m. Plugging these values into the formula, we get v = √(2 * 9.8 m/s^2 * 0.8 m) = 3.136 m/s. Now we can calculate the final angular velocity, ω = 3.136 m/s / 0.8 m = 3.92 rad/s.

Step 4: Calculate the final kinetic energy
Now that we have all the necessary values, we can plug them into the formula for kinetic energy, K = 1/2 * I *