Question

A steel cylinder with radius \(2.50 \mathrm{~cm}\) and length \(10.0 \mathrm{~cm}\) rolls without slipping down a ramp that is inclined at \(15.0^{\circ}\) above the horizontal and has a length (along the ramp) of \(3.00 \mathrm{~m}\). What is the induced potential difference between the ends of the cylinder as the cylinder leaves the bottom of the ramp, if the downward slope of the ramp points in the direction of the Earth's magnetic field at that location? (Use \(0.426 \mathrm{G}\) for the local strength of the Earth's magnetic field.)

Short answer

Solution: 1. To find the induced potential difference, we first need to find the angular velocity of the cylinder: - Calculate the height of the ramp: \(h = 3.00 \sin 15.0^{\circ} \) - Determine the potential energy at the top of the ramp: \(PE = mgh\) - Use the conservation of energy principle to find the final velocity of the cylinder at the bottom of the ramp. - Substitute the calculated values into the kinetic energy formula and find the linear velocity. 2. Now, using the linear velocity, find the angular velocity of the cylinder: - Use the relation between linear and angular velocities: \(\omega = \frac{v}{r}\) 3. Finally, calculate the induced potential difference: - Convert the magnetic field from G to T: \(0.426 \mathrm{G} = 0.426 \times 10^{-4} \mathrm{T}\) - Use Faraday's Law formula for a spinning disk: \(V = \frac{1}{2}\omega B r^2\) - Substitute the calculated values for \(\omega\), \(B\), and \(r\) to find the induced potential difference. With the given data and steps, we can find the induced potential difference between the ends of the cylinder as it leaves the bottom of the ramp.

Step-by-step solution

1. Find the final velocity of the cylinder at the bottom of the ramp

We can use the conservation of mechanical energy principle to find the final velocity of the cylinder. The potential energy at the top of the ramp is converted into kinetic energy at the bottom of the ramp. Let's compute the height of the ramp using trigonometry. The height of the ramp is \(h = L \sin\theta\) where \(L = 3.00\mathrm{~m}\) is the length of the ramp, and \(\theta = 15.0^{\circ}\) is the angle of the ramp above the horizontal. Thus, we have: \(h = 3.00 \sin 15.0^{\circ}\). Now, find the potential energy (\(PE\)) of the cylinder at the top of the ramp: \(PE = mgh\) where \(m\) is the mass of the cylinder, \(g \approx 9.81 \mathrm{~m/s^2}\) is the acceleration due to gravity, and \(h\) is the height (as found above). When the cylinder rolls without slipping, its kinetic energy at the bottom of the ramp is given by \(KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\) where \(v\) is the linear velocity of the cylinder, \(I = \frac{1}{2}mr^2\) is the moment of inertia of a solid cylinder (with \(r\) being the radius), and \(\omega\) is the angular velocity. Since the cylinder rolls without slipping, the linear and angular velocities are related by \(v = r\omega\). Using the conservation of energy principle \((PE = KE)\), we can find the final velocity (\(v\)) of the cylinder at the bottom of the ramp.

2. Calculate the angular velocity of the cylinder

Now that we have derived the linear velocity (\(v\)) of the cylinder in terms of \(m\), \(g\), and \(h\), we can find the angular velocity (\(\omega\)) using the relation \(v = r\omega\), where \(r = 2.50\mathrm{~cm}\) is the radius of the cylinder: \(\omega = \frac{v}{r}\)

3. Calculate the induced potential difference

With the angular velocity (\(\omega\)) found in the previous step, we can now find the induced potential difference (\(V\)) between the ends of the cylinder using Faraday's Law of electromagnetic induction and considering the cylinder as a spinning disk in a magnetic field. The magnetic field (\(B\)) is given as \(0.426 \mathrm{G} = 0.426 \times 10^{-4} \mathrm{T}\). The induced potential difference is given by: \(V = \frac{1}{2}\omega B r^2\) where \(\omega\) is the angular velocity, \(B\) is the strength of the magnetic field, and \(r\) is the radius of the cylinder. Substitute the calculated values for \(\omega\), \(B\), and \(r\), and find the induced potential difference (\(V\)) between the ends of the cylinder as it leaves the bottom of the ramp.