Question

A cylinder placed in frictionless bearings is set rotating about its axis. The cylinder is then heated, without mechanical contact, until its radius is increased by \(0.18 \%\). What is the percent change in the cylinder's ( \(a\) ) angular momentum, \((b)\) angular velocity, and ( \(c\) ) rotational energy?

Short answer

The percent change in the cylinder's angular momentum is 0%, in angular velocity is -0.36%, and in rotational energy is -0.72%.

Step-by-step solution

Determine the initial moment of inertia, angular momentum and rotational energy

Initially, let the energy, angular momentum and moment of inertia be E, L, and I respectively with the radius r. Thus, \( I = 0.5 m r^2 \), the angular momentum \( L = I w = 0.5 m r^2 w \), and the rotational energy \( E = 0.5 I w^2 = 0.25 m r^2 w^2 \). Note: All quantities are initial quantities.

Determine the final moment of inertia, angular momentum and rotational energy

After heating, the radius increases by 0.18%, which means the new radius r' is \( 1.0018 r \). The moment of inertia is now \( I' = 0.5 m (r')^2 = 0.5 m (1.0018 r)^2 \). Since there are no external torques, the angular momentum must be conserved, so we have \( L = I' w' \) where \( w' \) is the new angular velocity and rearrange to find \( w' = L / I' = 0.5 m r^2 w / 0.5 m (1.0018 r)^2 = w / 1.0018^2 \). The new rotational energy is \( E' = 0.5 I' (w')^2 = 0.25 m (1.0018 r)^2 (w / 1.0018^2)^2 \).

Compute the percentage changes

The percentage change in angular momentum is \((L' - L) / L * 100% = (L - L) / L * 100% = 0% \) since angular momentum is conserved. The percentage change in angular velocity is \((w' - w) / w * 100% = (w / 1.0018^2 - w) / w * 100% = -0.36% \). The percentage change in rotational energy is \( (E' - E) / E * 100% = (0.25 m (1.0018 r)^2 (w / 1.0018^2)^2 - 0.25 m r^2 w^2) / 0.25 m r^2 w^2 * 100% = -0.72% \).

Key concepts

Rotational Dynamics

Rotational dynamics is a branch of physics that describes the motion of rotating objects. In this scenario, the cylinder is allowed to rotate around its axis. The key aspect to understand in rotational dynamics is the concept of angular momentum. Angular momentum is a measure of an object's rotational motion and is conserved in a closed system where no external torques act upon it. It is given by the equation \( L = I\omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

In this specific exercise, the moment of inertia \( I \) for the cylinder is initially defined as \( I = 0.5mr^2 \), where \( m \) is the mass of the cylinder, and \( r \) is its radius. Despite the increase in the cylinder's radius due to heating, the absence of external forces means the angular momentum \( L \) remains constant. With no change in \( L \), the changes arise in the cylinder's angular velocity and rotational energy.

This exploration of rotational dynamics via a heated cylinder provides insight into how physical changes in a system affect its rotational properties while still maintaining conservation laws.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change its dimensions in response to temperature changes. When the cylinder is heated, its materials expand, leading to an increase in its radius. In our problem, the radius increases by \( 0.18\% \). This small relative change indicates a linear expansion, typical of solid objects under uniform temperature change.

The new radius can be calculated as \( r' = 1.0018r \), reflecting the increase due to thermal expansion. This change in radius affects the moment of inertia, which in turn affects rotational dynamics. Specifically, since the moment of inertia depends on \( r^2 \), an increase in radius results in a quadratic increase, impacting the object's ability to maintain angular velocity. Recognizing these physical changes is essential in applications of rotational dynamics and thermal effects.

Percentage Change Calculation

Calculating percentage change is a fundamental skill in many physics problems, including this one. Here, it's used to quantify the effects of heating the cylinder on its rotational properties.

In the exercise, the percentage changes were calculated for angular momentum, angular velocity, and rotational energy:- Angular momentum remained constant, thus a \(0\%\) change.- Angular velocity showed a \(-0.36\%\) change.- Rotational energy decreased by \(-0.72\%\).

To calculate these percentage changes, the following formula is applied: \[ \text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% \]For example, substitution in this equation allows derivation of results by comparing the final values after thermal expansion with their initial states. These calculations provide crucial insights into how physical changes impact a system's behavior and adhere to conservation laws.