Question

A turntable of mass \(5.00 \mathrm{kg}\) has a radius of \(0.100 \mathrm{m}\) and spins with a frequency of 0.550 rev/s. What is its angular momentum? Assume the turntable is a uniform disk.

Short answer

Answer: The angular momentum of the turntable is approximately 0.0864 kg·m²/s.

Step-by-step solution

Find the moment of inertia of the uniform disk

To find the moment of inertia, we will use the formula for the moment of inertia of a uniform disk, which is given by: \(I=\frac{1}{2}MR^2\), where M is the mass and R is the radius of the disk. In our case, the mass is 5.00 kg and radius is 0.100 m, so plug the values into the formula.\(I=\frac{1}{2}(5.00\,\text{kg})(0.100\,\text{m})^2\)

Calculate the moment of inertia

Now, we can simplify the formula to find the moment of inertia.\(I=\frac{1}{2}(5.00\,\text{kg})(0.0100\,\text{m}^2) = 0.0250\,\text{kg}\cdot\text{m}^2\)

Convert the frequency to the angular velocity

Since we are given the frequency in rev/s, we'll need to convert it to angular velocity in rad/s. The conversion factor is \(2\pi\) radians per revolution because there are \(2\pi\) radians in one revolution. Therefore, multiply the given frequency by \(2\pi\) to find the angular velocity. \(\omega = 0.550\,\text{rev/s}\times 2\pi\,\text{rad/rev}\)

Calculate the angular velocity

Now, we can simplify the formula to find the angular velocity. \(\omega = 0.550\,\text{rev/s}\times 2\pi\,\text{rad/rev} \approx 3.455\,\text{rad/s}\)

Find the angular momentum

Now we have everything we need to find the angular momentum, L. We'll use the formula \(L = I\omega\), where I is the moment of inertia, and \(\omega\) is the angular velocity. Plug in the values we calculated in Step 2 and Step 4: \(L = (0.0250\,\text{kg}\cdot\text{m}^2)(3.455\,\text{rad/s})\)

Calculate the angular momentum

Now, we'll simply multiply the moment of inertia and the angular velocity to get the final answer for the angular momentum. \(L = (0.0250\,\text{kg}\cdot\text{m}^2)(3.455\,\text{rad/s}) \approx 0.0864\,\text{kg}\cdot\text{m}^2\text{/s}\) The angular momentum of the turntable is approximately 0.0864 kg·m²/s.