Problem 56 - College Physics

Question

A uniform sphere and a uniform cylinder with the same mass and radius roll at the same velocity side by side on a level surface without slipping. If the sphere and the cylinder approach an inclined plane and roll up it without slipping, will they be at the same height on the plane when they come to a stop? If not, what will be the percentage difference of the heights?

Step-by-Step Solution

Answer

No, the sphere and cylinder won't reach the same height; the cylinder will reach 6.25% less height than the sphere.

Step 1: Identifying Kinetic Energy Components

Both the sphere and the cylinder possess translational kinetic energy due to their linear motion and rotational kinetic energy due to their rotation. The translational kinetic energy (TKE) for each object is given by \( \frac{1}{2} m v^2 \), where \( m \) is mass and \( v \) is velocity. The rotational kinetic energy (RKE) for each is \( \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is angular velocity.

Step 2: Moment of Inertia and Angular Velocity

The moment of inertia for a solid sphere is \( I = \frac{2}{5}mr^2 \), and for a solid cylinder, \( I = \frac{1}{2}mr^2 \). The relation between angular velocity \( \omega \) and linear velocity \( v \) is given as \( v = r\omega \), so \( \omega = \frac{v}{r} \) for both objects.

Step 3: Calculating Total Energy for Each Object

For each object, the total kinetic energy (TKE + RKE) is calculated. For the sphere: \( \text{Total Energy} = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{5}mr^2)(\frac{v}{r})^2 = \frac{7}{10}mv^2 \). For the cylinder, \( \text{Total Energy} = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{1}{2}mr^2)(\frac{v}{r})^2 = \frac{3}{4}mv^2 \).

Step 4: Conservation of Energy as They Roll Up the Incline

As both roll up the incline without slipping, their kinetic energy converts entirely into potential energy (PE) at the maximum height. Potential energy at height \( h \) is \( mgh \), where \( g \) is the acceleration due to gravity. Set \( mgh \) equal to the total kinetic energy from Step 3 to find height \( h \).

Step 5: Calculating Maximum Heights

For the sphere, \( mgh_s = \frac{7}{10}mv^2 \) gives \( h_s = \frac{7v^2}{10g} \). For the cylinder, \( mgh_c = \frac{3}{4}mv^2 \) gives \( h_c = \frac{3v^2}{4g} \).

Step 6: Determine Percentage Difference in Heights

Calculate the percentage difference using the formula: \( \text{Percentage Difference} = \left( \frac{h_c - h_s}{h_s} \right) \times 100 \% \). Substitute and simplify: \( \text{Percentage Difference} = \left( \frac{\frac{3}{4} - \frac{7}{10}}{\frac{7}{10}} \right) \times 100 \% = 6.25 \% \).

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, which can be both translational and rotational. For objects like a sphere or a cylinder moving along a surface, we need to account for both these forms.

• Translational kinetic energy is given by the formula: \( \frac{1}{2} m v^2 \), where \( m \) is the mass and \( v \) is the velocity of the object. This formula represents the energy due to the object moving along a path.
• Rotational kinetic energy, on the other hand, depends on how the object spins. It is calculated as \( \frac{1}{2} I \omega^2 \), where \( I \) represents the moment of inertia, and \( \omega \) is the angular velocity.
  

By understanding both forms, we see that a rolling object combines these energies. This combination must be considered to understand how objects behave when they encounter frictionless inclined planes.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the object's mass distribution relative to the axis of rotation.

• A solid sphere, which is more compact, has a moment of inertia given by \( I = \frac{2}{5}mr^2 \).
• A solid cylinder, which spreads its mass further from the axis, has a moment of inertia \( I = \frac{1}{2}mr^2 \).

The lower the moment of inertia, the easier it is to change the object's rotational state. Thus, understanding the moment of inertia helps in predicting how fast a body will rotate for a given amount of applied energy.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In our scenario with the sphere and cylinder, kinetic energy is converted to potential energy as they roll up the incline.

• When an object rolls up an incline, its kinetic energy decreases while its potential energy increases.
• The total energy remains constant throughout the motion as long as no external forces (like friction or air resistance) do significant work on the objects.

By setting kinetic energy equal to potential energy at the maximum height, we determine how high each object can climb on the incline.

Rotational Motion

Rotational motion describes the movement of an object around a central axis. Understanding the connection between linear and rotational motion is crucial in physics.

• The relationship between linear velocity \( v \) and angular velocity \( \omega \) is expressed as \( v = r \omega \). For rolling without slipping, this holds true for both spherical and cylindrical objects.
• This equation allows us to express everything in terms of linear velocity, which makes the analysis of kinetic energy cleaner when considering objects that roll.

Understanding these principles helps in modeling systems that have parts moving along curved paths or rotating about pivots.

Percentage Difference Calculation

Percentage difference calculations help compare differences in quantities relative to their sizes, a useful tool in physics.
To find the percentage difference between two heights, we use the formula:\[ \text{Percentage Difference} = \left( \frac{h_c - h_s}{h_s} \right) \times 100 \% \]Here:

• \( h_c \) is the height for the cylinder, and
• \( h_s \) is the height for the sphere.

Substituting the values, the percentage difference is calculated as 6.25%. This tells us how much more, in percentage terms, one object climbs in comparison to another.