Problem 66 - College Physics

Question

A solid uniform spherical stone starts moving from rest at the top of a hill. At the bottom of the hill, the ground curves upward, launching the stone vertically a distance \(H\) below its start. How high will the stone go (a) if there is no friction on the hill and (b) if there is enough friction on the hill for the stone to roll without slipping? (c) Why do you get two different answers even though the stone starts with the same gravitational potential energy in both cases?

Step-by-Step Solution

Answer

(a) Height = H. (b) Height = \(\frac{5}{7}H\). (c) Rotational energy reduces height in case (b).

Step 1: Identify the Energy Conversion

Initially, the stone has gravitational potential energy at height \(H\). As it moves downward, this energy is converted into kinetic energy. At the top of its trajectory after being launched, all the energy will again be potential energy.

Step 2: Scenario (a) - Sliding Without Friction

In the scenario without friction, all gravitational potential energy is converted into translational kinetic energy. Using the energy conservation principle: \(mgh = \frac{1}{2}mv^2\). Solving for final height \(h_f\), we have \( h_f = H \). So in this case, the stone reaches the height \( H \).

Step 3: Scenario (b) - Rolling Without Slipping

When the stone rolls without slipping, the potential energy converts into both translational and rotational kinetic energies. Using energy conservation: \(mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\). Substitute \( I = \frac{2}{5}mr^2 \) and \( \omega = v/r \), solve for \( h_f \); the stone reaches \( \frac{5}{7}H \).

Step 4: Explain Different Heights in (c)

In scenario (a), all potential energy is converted into translational kinetic energy. In scenario (b), some potential energy also goes into rotational motion, reducing energy available for vertical displacement. Consequently, the stone climbs to a lower height in case (b).

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position relative to the Earth, or any other gravitational field. This type of energy depends on the height of the object above a reference point, often taken to be the ground. It is given by the formula \(E_{p} = mgh\), where:

• \(E_{p}\) is the gravitational potential energy,
• \(m\) is the mass of the object,
• \(g\) is the acceleration due to gravity (approximately \(9.81 \ m/s^2\) on Earth), and
• \(h\) is the height above the chosen reference point.

As the stone begins its descent from the top of the hill, it starts with maximum gravitational potential energy. As it moves down, this energy gets converted into other forms, primarily kinetic energy.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is determined by the object's mass and velocity, expressed in the formula \(E_{k} = \frac{1}{2}mv^2\). The faster an object moves, the more kinetic energy it has.

• \(E_{k}\) is the kinetic energy,
• \(m\) is the mass, and
• \(v\) is the velocity of the object.

In the case of the stone rolling down the hill, as it accelerates, its velocity increases, converting its initial gravitational potential energy into kinetic energy. Without friction, this conversion is entirely into translational kinetic energy, which governs how quickly the stone moves. With friction, some energy is diverted into rotational motion as well.

Rotational Motion

Rotational motion involves spinning around an axis. When a stone rolls down a slope, rotational motion is introduced, meaning that the stone not only moves forward but also spins. For rolling motion without slipping, both translational and rotational motion occur together.The rotational kinetic energy is given by \(E_{r} = \frac{1}{2}I\omega^2\), where:

• \(E_{r}\) is the rotational kinetic energy,
• \(I\) is the moment of inertia, which is \(\frac{2}{5}mr^2\) for a solid sphere, and
• \(\omega\) is the angular velocity, related to the linear velocity \(v\) by \(\omega = \frac{v}{r}\).

In the scenario where the stone rolls without slipping, some portion of the gravitational potential energy goes into causing the stone to rotate, reducing the amount converted into linear motion, and affecting the height it reaches after being launched.

Energy Conversion

Energy conversion is a fundamental concept where energy changes from one form to another while maintaining the total energy. In the stone's journey from the top to the bottom of the hill, gravitational potential energy converts to kinetic energy. If no friction is present, all energy converts into translational kinetic energy. However, when friction is involved, as in rolling without slipping, the energy conversion involves two steps:

• Gravitational potential energy to translational kinetic energy, and
• Gravitational potential energy to rotational kinetic energy.

This is why, in the rolling scenario, the stone does not reach as high a point upon launching because some energy is used to maintain the motion of spinning, illustrating conservation of energy and the impact of energy partitioning in different forms.