Problem 83 - College Physics

Question

A uniform sphere of mass \(2.50 \mathrm{~kg}\) and radius \(15.0 \mathrm{~cm}\) is released from rest at the top of an incline that is \(5.25 \mathrm{~m}\) long and makes an angle of \(35^{\circ}\) with the horizontal. Assuming it rolls without slipping, (a) determine its total kinetic energy at the bottom of the incline. (b) Determine its rotational kinetic energy at the bottom of the incline. (c) What type of friction, static or kinetic, is acting on the surface of the sphere? Explain. (d) Determine the force of friction in part (d).

Step-by-Step Solution

Answer

(a) Use potential energy to find total kinetic energy at bottom. (b) Rotational kinetic is \(\frac{2}{7}\) of total kinetic energy. (c) Static friction is at play. (d) Friction force is \(\frac{2}{5}mg\sin(\theta)\).

Step 1: Calculate gravitational potential energy at the top

At the top of the incline, the total potential energy is due to gravity. The height of the incline can be determined using trigonometry: \( h = 5.25 \sin(35^\circ) \). The potential energy \( PE \) is:\[PE = mgh = (2.50 \text{ kg})(9.81 \text{ m/s}^2)(5.25 \sin(35^\circ))\]

Step 2: Find total kinetic energy at the bottom using energy conservation

Assuming no energy loss, the potential energy at the top converts entirely into kinetic energy at the bottom. Thus, the total kinetic energy \( KE_{total} \) is equal to the potential energy calculated:\[KE_{total} = PE\]

Step 3: Express total kinetic energy as a sum of translational and rotational energy

The total kinetic energy is the sum of translational and rotational kinetic energy:\[KE_{total} = KE_{trans} + KE_{rot}\]Where \( KE_{trans} = \frac{1}{2}mv^2 \) and \( KE_{rot} = \frac{1}{2}I\omega^2 \). The moment of inertia \( I \) for a sphere is \( \frac{2}{5}mr^2 \). The relation between linear and angular velocity is \( v = r\omega \).

Step 4: Calculate rotational kinetic energy

Using the relation \( v = r\omega \) and substituting \( \omega = \frac{v}{r} \) into the rotational kinetic energy formula:\[KE_{rot} = \frac{1}{2} \left(\frac{2}{5}mr^2\right)\left(\frac{v}{r}\right)^2 = \frac{1}{5}mv^2\]Since \( KE_{total} = \frac{7}{10}mv^2 \), solve for \( KE_{rot} \) using the energy conservation equation:\[KE_{rot} = \frac{2}{7}KE_{total}\]

Step 5: Identify the type of friction

The sphere rolls without slipping, which implies that static friction is at play. Static friction acts to prevent slipping, providing the necessary torque for rolling.

Step 6: Calculate the force of friction

The force of static friction can be found using Newton's second law for rotational motion:\[\tau = I\alpha = f_s r\]Since \( \alpha = \frac{a}{r} \) and the linear acceleration \( a \) is given by \( a = g \sin(\theta) \), solve for the friction force \( f_s \):\[f_s = \frac{2}{5}ma = \frac{2}{5}mg\sin(\theta)\]

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. For the sphere rolling down the incline, it involves both translational and rotational motion.

• Translational kinetic energy refers to the energy of the sphere's center of mass moving in a straight line. It is calculated as \( KE_{trans} = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.
• Rotational kinetic energy, on the other hand, accounts for the energy due to rotation around its axis. This is given by \( KE_{rot} = \frac{1}{2}I\omega^2 \), with \( I \) as the moment of inertia and \( \omega \) as the angular velocity.

The sphere's total kinetic energy at the bottom of the incline is the sum of these two energies. It shows how energy transforms from potential to kinetic as the sphere descends.

Potential Energy

Potential energy is the stored energy of an object due to its position in a force field, commonly gravitational. At the top of the incline, the sphere has maximum potential energy, given by the formula: \[PE = mgh\]where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height of the sphere above the ground.

• This height is determined by the length of the incline and the angle it makes with the horizontal, which can be found using trigonometry \( h = 5.25\sin(35^\circ) \).
• The initial potential energy converts into kinetic energy as the sphere descends.

Understanding potential energy helps to explain how energy is conserved as the sphere moves.

Static Friction

Static friction is the force that prevents surfaces from sliding past each other. It plays a crucial role in the sphere's motion as it rolls down without slipping. For rolling to occur without slipping:

• Static friction provides the necessary torque to rotate the sphere.
• It ensures that the bottom point of the sphere remains momentarily at rest with the surface.

In this exercise, static friction does not do work on the sphere since there is no displacement at the point of contact. Instead, it allows the energy to remain within the system as the sphere rolls smoothly.

Energy Conservation

Energy conservation is a fundamental principle stating that the total energy in an isolated system remains constant. In the scenario of the rolling sphere, this means:

• The gravitational potential energy at the top entirely transforms into kinetic energy at the bottom.
• No energy is lost through non-conservative forces, assuming the absence of air resistance and frictional heating.

This process illustrates the conversion between potential and kinetic energies. The calculation shows how potential energy decreases as the sphere moves downhill, while kinetic energy increases, keeping the total energy consistent. Understanding energy conservation is essential for solving dynamics problems effectively.