Consider a spinning plate is dropped onto a stationary plate (which is itself at rest on a frictionless surface). Both plates have a radius of 30cm and a mass of 1kg. The spinning plate is initially spinning at a rate of 0.7 revolutions per second. Hint: This is like a totally-inelastic collision. Required: a. After a sufficiently long time, what is the angular velocity of the initially-spinning plate? What about the initially-stationary plate? b. Assume that the period of velocity matching happens over a course of 2 seconds. Further, assume that the torque exerted by each plate on the other is constant over time. In that case, what is the magnitude of the acceleration that each plate feels during those two seconds? Hint: Use the rotational impulse-momentum theorem.

a. When the spinning plate is dropped onto the stationary plate, since the surface is frictionless and assuming that there is no other external torque acting on the system, the angular momentum of the system is conserved. This situation is analogous to a totally inelastic collision in linear motion but in the context of rotational motion.

Since the plates stick together, they will eventually spin with the same angular velocity. Let's call the final angular velocity of both plates together as 'ω'.

Before the collision, the spinning plate has angular momentum, which can be calculated using L = I * ω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. Because both plates are identical, they have the same moment of inertia. The initial angular momentum of the system is just the angular momentum of the spinning plate because the stationary plate has no angular momentum (it isn't moving).

The moment of inertia of a single disk (plate) about its center is I = (1/2) * m * r^2, where m is the mass and r is the radius.

Let's calculate the initial moment of inertia of the spinning plate:

I_plate = (1/2) * 1 kg * (0.3 m)^2 = (1/2) * 1 * 0.09 = 0.045 kg·m^2

The angular velocity in revolutions per second needs to be converted to radians per second (since there are 2π radians in one revolution):

ω_initial = 0.7 rev/s * 2π rad/rev = 1.4π rad/s

So, the initial angular momentum of the system is:

L_initial = I_plate * ω_initial = 0.045 kg·m^2 * 1.4π rad/s = 0.063π kg·m^2/s

Conservation of angular momentum demands that:

L_initial = L_final

Since both plates will spin together after they collide and stick, their combined moment of inertia is:

I_combined = I_plate + I_plate = 2 * I_plate = 2 * 0.045 kg·m^2 = 0.09 kg·m^2

So, using the conservation of angular momentum:

L_initial = I_combined * ω final

Let's solve for the final angular velocity (ω final):

0.063π kg·m^2/s = 0.09 kg·m^2 * ω_final ω_final = (0.063π kg·m^2/s) / (0.09 kg·m^2) ω_final ≈ 0.7π rad/s

Thus, both the initially-spinning plate and the initially-stationary plate will eventually spin with an angular velocity of approximately 0.7π rad/s after a sufficiently long time.

b. According to the rotational impulse-momentum theorem, the change in angular momentum is equal to the rotational impulse, which is the product of torque and the time over which the torque acts.

ΔL = torque * Δt

Both plates are subjected to an equal and opposite torque due to Newton's third law, and this results in an equal and opposite change in their angular momenta over the 2-second interval.

But first, let's find ΔL for one of the plates. Since we know L_final and L_initial for the spinning plate, we can calculate ΔL:

ΔL_spinning_plate = L_final - L_initial = (I_plate * ω_final) - (I_plate * ω_initial) = 0.045 kg·m^2 * (0.7π rad/s - 1.4π rad/s) = 0.045 kg·m^2 * -0.7π rad/s = -0.0315π kg·m^2/s

The minus sign indicates that the spinning plate's angular momentum decreases.

The torque τ experienced by each plate can be found using the rotational impulse-momentum theorem over the 2-second interval:

τ * Δt = ΔL τ * 2 s = -0.0315π kg·m^2/s τ = (-0.0315π kg·m^2/s) / 2 s τ ≈ -0.01575π N·m

The negative sign indicates the direction of the torque relative to the orientation we would assign to a positive angular acceleration.

Now to find the angular acceleration α, we use Newton's second law for rotation:

τ = I * α

We calculate α for each plate (note that I here is that of a single plate):

α = τ / I_plate α ≈ (-0.01575π N·m) / (0.045 kg·m^2) α ≈ -0.35π rad/s^2

Again, the negative sign reflects the direction relative to our positive torque orientation. Both plates experience an angular acceleration of magnitude approximately 0.35π rad/s^2 during the 2 seconds that it takes for their velocities to match.

Extra: When dealing with rotational motion, it's important to understand the analogy between linear and rotational dynamics. In linear motion, we consider quantities like mass, velocity, momentum, and forces, whereas in rotational motion, the analogous quantities are the moment of inertia, angular velocity, angular momentum, and torque.

Moment of inertia (I) is the rotational equivalent to mass in linear motion; it describes an object's resistance to changes in its rotational motion. Angular velocity (ω) is similar to linear velocity but for rotation, measured in radians per second. Angular momentum (L) is the product of moment of inertia and angular velocity and describes the quantity of rotation an object has, analogous to linear momentum in translational motion. Torque (τ) is the rotational equivalent of force, causing an object to rotate about an axis.

Conservation of angular momentum is a fundamental principle that states that if there are no external torques on a system, the total angular momentum before an event (such as a collision) must equal the total angular momentum after the event. This principle allows us to solve problems involving rotational collisions, like the one with the plates.

Finally, the rotational impulse-momentum theorem is the rotational analogy to the impulse-momentum theorem in linear dynamics. It relates the torque applied to an object over a period of time with the change in its angular momentum. This theorem lets us solve for unknowns such as final angular velocities or the torques involved when objects interact in a rotational context.