Two identical wheels are moving on horizontal surfaces. The center of mass of each has the same linear speed. However, one wheel is rolling, while the other is sliding on a frictionless surface without rolling. Each wheel then encounters an incline plane. One continues to roll up the incline, while the other continues to slide up. Eventually they come to a momentary halt, because the gravitational force slows them down. Each wheel is a disk of mass 2.02 kg. On the horizontal surfaces the center of mass of each wheel moves with a linear speed of 6.12 m/s. (a),(b) What is the total kinetic energy of each wheel? (c), (d) Determine the maximum height reached by each wheel as it moves up the incline.
One step that I am confused about is finding the KEr as the formula shown in the tutorial is (3*m*v^2)/4. May I ask how this is found?
The formula you are referring to, (3*m*v^2)/4, is the expression for the rotational kinetic energy (KEr) of a solid disk rolling without slipping. It is derived based on the relationship between the linear speed (v) and rotational speed (ω) of a rolling object, as well as the moment of inertia (I) of the object.
To understand how this formula is obtained, let's break it down step by step:
1. First, we need to know the moment of inertia (I) for a solid disk. The moment of inertia is a measure of how mass is distributed around an axis of rotation. For a solid disk rotating about its central axis, the moment of inertia is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the disk and r is its radius.
2. Next, we need to relate the linear speed (v) of the disk to its rotational speed (ω). For a disk rolling without slipping, the linear speed at the edge of the disk is equal to the radius times the rotational speed:
v = r * ω
Rearranging this equation, we can solve for ω:
ω = v / r
3. Now, let's go back to the formula for rotational kinetic energy (KEr):
KEr = (1/2) * I * ω^2
Substituting the value of I from step 1, and ω from step 2, we get:
KEr = (1/2) * (1/2) * m * r^2 * (v^2 / r^2)
Simplifying, we find:
KEr = (1/4) * m * v^2
This is the formula you mentioned: (3*m*v^2)/4. However, there seems to be a mistake in the formula you provided. The correct formula for the rotational kinetic energy of a solid disk rolling without slipping is (1/4) * m * v^2, not (3*m*v^2)/4. So, please use the corrected formula.
I hope this explanation helps you understand how the formula for rotational kinetic energy of a rolling disk is derived. Let me know if you have any further questions!