Problem 96 - University Physics with Modern Physics

Question

A doubling of the torque produces a greater angular acceleration. Which of the following would do this, assuming that the tension in the rope doesn't change? (a) Increasing the pulley diameter by a factor of \(\sqrt{ 2 }\); (b) increasing the pulley diameter by a factor of 2; (c) increasing the pulley diameter by a factor of 4; (d) decreasing the pulley diameter by a factor of \(\sqrt{ 2 }\).

Step-by-Step Solution

Answer

Option (b) increases the pulley diameter by a factor of 2, doubling the torque.

Step 1: Understanding Torque and Angular Acceleration

Torque \( \tau \) is related to angular acceleration \( \alpha \) by the equation \( \tau = I \alpha \), where \( I \) is the moment of inertia of the rotating object. If the torque doubles, the angular acceleration will also double.

Step 2: Relation Between Torque, Tension, and Radius

Torque is also given by \( \tau = r \cdot T \), where \( r \) is the radius of the pulley and \( T \) is the tension in the rope. Here, the tension remains constant, hence any change in torque must come from changing the radius, \( r \).

Step 3: Calculating Influence of Pulley Diameter Changes

Increasing the pulley diameter increases its radius, while decreasing the diameter decreases its radius. For the torque to double (2\( \tau \)), the radius must also double, since \( T \) is constant. Thus, we need \( 2r = r_{new} \). Based on the options, we will test which increase in radius reflects a doubling.

Step 4: Evaluate Each Option Against Torque Criteria

- Option (a): Increasing the diameter by a factor of \( \sqrt{2} \) changes the radius by a factor of \( \sqrt{2} \), which results in \( r_{new} = \sqrt{2}r \). This does not double the torque.- Option (b): The radius increases by a factor of 2, making \( r_{new} = 2r \). This correctly doubles the torque.- Option (c): The radius increases by a factor of 4, making \( r_{new} = 4r \). This more than doubles the torque.- Option (d): Decreasing the radius by \( \sqrt{2} \) results in \( r_{new} = r/\sqrt{2} \), which reduces, not increases, the torque.

Key Concepts

Torque

Torque is a fundamental concept in rotational dynamics. It can be thought of as the rotational equivalent of force. When a force causes an object to rotate, it is applying torque. In our scenario, torque is given by the simple formula: \( \tau = I \alpha \). Here, \(I\) represents the moment of inertia, and \(\alpha\) is the angular acceleration. This equation shows that, given a constant moment of inertia, an increase in torque will lead to a proportional increase in angular acceleration.

Torque also links with the radius and tension, as seen in the formula: \(\tau = r \cdot T\). The radius \(r\) is the distance from the axis of rotation to the point where the force (or tension, \(T\)) is applied. Thus, the torque produced depends on both the magnitude of the force and where it is applied relative to the axis.

Understanding these relationships can help solve many problems involving rotational motion, making torque a key concept for students of physics.

Moment of Inertia

The moment of inertia, denoted as \(I\), measures an object's resistance to changes in its rotational motion. It is dependent on how an object's mass is distributed relative to the axis of rotation. Just as mass resists changes in linear motion, the moment of inertia resists changes in rotational motion.

In our exercise, the moment of inertia is involved in the formula \(\tau = I \alpha\). If the moment of inertia is constant, doubling the torque will result in doubling the angular acceleration \(\alpha\).

The moment of inertia is crucial when analyzing systems such as pulleys or rotating discs where distribution of mass affects motion. In practical terms, increasing the size of the pulley changes its moment of inertia, which can influence how effectively it rotates under a given torque.

Pulley Diameter

Pulley diameter affects both the radius \(r\) and the moment of inertia \(I\) of the pulley. When we vary the diameter, we directly change the radius: if the diameter increases, the radius increases proportionally. Since torque \(\tau = r \cdot T\) relies on this radius, altering the diameter can change the torque.

Consider our exercise: with constant tension, to double the torque, the radius must double. This means increasing the pulley diameter by a factor of 2 effectively achieves the desired torque. Other options either do not fully double or vastly exceed the required increase.

Adjusting the diameter also affects the moment of inertia. Larger diameters mean a larger moment of inertia, which influences how the system reacts to torque changes. Understanding how diameter influences these factors is fundamental when solving mechanical problems involving pulleys.

Tension in Rope

Tension in the rope, denoted as \(T\), contributes to the torque developed in a pulley system. In the context of our exercise, the torque is computed with the formula \(\tau = r \cdot T\). The tension remains unchanged, which means to alter torque, we adjust the radius, rather than the tension itself.

Tension is simply the force exerted by the rope. It is critical in ensuring that the applied force leads to effective rotational motion. As long as the tension is consistent, the primary variable controlling torque changes is the radius.

Thus, understanding the role of tension helps in comprehending how force is transmitted efficiently through a system. It also emphasizes why, in our exercise, changing the pulley’s radius was necessary to achieve the goal of doubling the torque, given the fixed tension.