Math

QuestionA student with mass mm runs and jumps onto the outer edge of a stationary horizontal platform that is free to rotate about an axis at the center of the platform. The platform has mass MM, radius RR, and rotational inertia 12MR2\frac{1}{2} M R^{2}. Immediately before landing on the platform the student has horizontal speed vv directed tangentially to the outer edge of the platform. Which of the following is equal to the angular momentum of the student about the platform's axis immediately before landing on the platform? (A) 12Rmv\frac{1}{2} R m v (B) Rmv (C) 12RMv\frac{1}{2} R M v (D) RMvR M v

Studdy Solution

STEP 1

What is this asking? Determine the angular momentum of a student running and jumping onto a rotating platform, just before they land. Watch out! Don't confuse linear momentum with angular momentum; they are related but distinct concepts!

STEP 2

1. Understand angular momentum
2. Calculate the angular momentum of the student

STEP 3

Alright, let's dive into angular momentum!
It's like the rotational equivalent of linear momentum.
For a point mass, like our student, angular momentum L L about a point is given by the formula:
L=rmvsin(θ)L = r \cdot m \cdot v \cdot \sin(\theta)where: - r r is the distance from the axis of rotation to the point mass, - m m is the mass of the object, - v v is the linear velocity, - θ\theta is the angle between the position vector and the velocity vector.

STEP 4

In this problem, since the student is running tangentially to the edge of the platform, θ=90\theta = 90^\circ, and sin(90)=1\sin(90^\circ) = 1.
This means the formula simplifies to:
L=rmvL = r \cdot m \cdot v

STEP 5

Now, let's plug in the values we know!
The student is at the edge of the platform, so the distance r r is equal to the radius R R of the platform.
The mass of the student is m m , and their speed is v v .
So, the angular momentum L L becomes:
L=RmvL = R \cdot m \cdot v

STEP 6

This is the angular momentum of the student about the platform's axis immediately before landing.
It's important to note that this calculation is independent of the platform's mass or rotational inertia, as we're only considering the student's contribution.

STEP 7

The angular momentum of the student about the platform's axis immediately before landing is Rmv\mathbf{Rmv}.

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