Torque on Discs & Pulleys: Videos & Practice Problems

Understanding torque is essential for analyzing the rotational motion of disc-like objects, such as pulleys and cylinders. The torque tau produced by a force depends on the distance (\( r \)) from the axis of rotation to the point where the force is applied, not merely the radius of the object. For example, when two masses are connected by a string around a cylinder, only the forces acting at the edges contribute to the net torque. The net torque can be calculated by summing the individual torques, considering their direction, leading to a comprehensive understanding of rotational dynamics.

Torque on Discs & Pulleys

Understanding torques acting on disc-like objects, such as pulleys or cylinders, is crucial for solving more complex problems in physics. When a force is applied to a disc, the distance from the axis of rotation to the point where the force is applied, denoted as r, plays a significant role in determining the torque produced. It is important to note that this distance is not always the same as the radius of the disc.

For example, consider a system with two masses, m₁ and m₂, connected by a light string that wraps around a solid cylinder. The cylinder has a mass m₃ and a radius R. When the blocks are released, the net torque on the cylinder can be calculated by summing the torques produced by the forces acting on it. The relevant forces include the gravitational forces acting on the masses and the weight of the cylinder itself.

The torque produced by a force is calculated using the formula:

Torque = F × r × sin(θ)

where F is the force, r is the distance from the axis of rotation to the point of force application, and θ is the angle between the force vector and the r vector. In this scenario, the forces due to the weights of the cylinder and the tension do not produce any torque because they act directly at the axis of rotation, resulting in an r of zero or an angle of zero degrees, both of which yield a torque of zero.

The only forces that contribute to the net torque are those acting at the edges of the disc, specifically m₁ and m₂. For m₁, the torque can be calculated as:

Torque₁ = m₁ × g × R × sin(90°) = m₁ × g × R

For m₂, the torque is similarly calculated:

Torque₂ = -(m₂ × g × R × sin(90°)) = -(m₂ × g × R)

In this case, if m₁ is 4 kg and m₂ is 5 kg, with g approximated to 10 m/s² and R as 3 m, the calculations yield:

Torque₁ = 4 kg × 10 m/s² × 3 m = 120 N·m (positive, counterclockwise)

Torque₂ = - (5 kg × 10 m/s² × 3 m) = -150 N·m (negative, clockwise)

Summing these torques gives the net torque:

Net Torque = 120 N·m - 150 N·m = -30 N·m

This negative value indicates that the net torque is in the clockwise direction. Understanding these principles of torque is essential for analyzing rotational motion and the dynamics of systems involving disc-like objects.

Net Torque on a disc

Calculating torque on a disc involves understanding how forces applied at various angles can influence its rotation. In this scenario, we have a composite disc consisting of an inner disc with a radius of 3 meters and an outer disc with a radius of 5 meters. The forces acting on the discs are all 100 newtons, with one force applied at a 37-degree angle, while the others are aligned either parallel or perpendicular to the axes.

Torque (\( \tau \)) is defined as the product of the force (\( F \)), the distance from the axis of rotation (\( r \)), and the sine of the angle (\( \theta \)) between the force vector and the lever arm. The formula for torque can be expressed as:

\[ \tau = F \cdot r \cdot \sin(\theta) \]

To find the net torque on the disc, we analyze each force individually:

1. **Torque from \( F_1 \)**: This force is applied at the edge of the outer disc, where \( r_1 = 5 \) meters and \( \theta_1 = 90^\circ \). The sine of 90 degrees is 1, leading to:

\[ \tau_1 = 100 \, \text{N} \cdot 5 \, \text{m} \cdot \sin(90^\circ) = 500 \, \text{N m} \] (positive, counterclockwise)

2. **Torque from \( F_2 \)**: This force acts directly on the axis of rotation, making \( r_2 = 0 \). Since torque depends on the distance from the axis, this results in:

\[ \tau_2 = 100 \, \text{N} \cdot 0 \cdot \sin(\theta) = 0 \, \text{N m} \]

3. **Torque from \( F_3 \)**: This force is also applied at the edge of the outer disc, but it acts in the opposite direction, with \( r_3 = 5 \) meters and \( \theta_3 = 180^\circ \). The sine of 180 degrees is 0, resulting in:

\[ \tau_3 = 100 \, \text{N} \cdot 5 \, \text{m} \cdot \sin(180^\circ) = 0 \, \text{N m} \]

4. **Torque from \( F_4 \)**: This force is applied at the inner disc with \( r_4 = 3 \) meters and at an angle of 37 degrees. The sine of 37 degrees is approximately 0.6018, leading to:

\[ \tau_4 = 100 \, \text{N} \cdot 3 \, \text{m} \cdot \sin(37^\circ) \approx 180.54 \, \text{N m} \] (positive, counterclockwise)

To find the net torque, we sum all the individual torques:

\[ \tau_{\text{net}} = \tau_1 + \tau_2 + \tau_3 + \tau_4 = 500 \, \text{N m} + 0 + 0 + 180.54 \, \text{N m} = 680.54 \, \text{N m} \]

Thus, the net torque acting on the disc is approximately 680.54 Newton meters in the counterclockwise direction. Understanding how to calculate torque from various forces is crucial for analyzing rotational motion in physics.

The composite disc below is free to rotate about a fixed axis, perpendicular to it and through its center. All forces are 100 N, and all angles are 37°. The dotted lines are either exactly parallel or exactly perpendicular to each other. The inner (darker) and outer (lighter) discs have radii 3 m and 5 m, respectively. Calculate the Net Torque produced on the composite disc, about an axis perpendicular to it and through its center. Use +/– to indicate direction.

• A. −100 N•m
• B. +100 N•m
• C. −800 N•m
• D. +800 N•m
• E. −900 N•m
• F. +900 N•m
• G. −1440 N•m
• H. +1440 N•m

Answer: (E)

A square with sides 4 m long is free to rotate around an axis perpendicular to its face and through its center. All forces shown are 100 N and act simultaneously on the square. The angle shown is 30°. Calculate the Net Torque that the forces produce on the square, about its axis of rotation.

• A. +207 N•m
• B. +327 N•m
• C. +348 N•m
• D. +473 N•m

Answer: (B)

Here’s what students ask on this topic:

What is the formula for calculating torque on a disc or pulley?

The formula for calculating torque (τ) on a disc or pulley is given by:

$\tau =Fr\sin \left(\theta \right)$

where:

• $F$ is the force applied.
• $r$ is the distance from the axis of rotation to the point where the force is applied.
• $\theta$ is the angle between the force vector and the lever arm.

This formula helps in understanding how different forces and their points of application affect the rotational motion of the disc or pulley.

How do you determine the net torque on a pulley system with multiple forces?

To determine the net torque on a pulley system with multiple forces, follow these steps:

1. Identify all the forces acting on the pulley.
2. Calculate the torque produced by each force using the formula $\tau =Fr\sin \left(\theta \right)$.
3. Determine the direction of each torque (clockwise or counterclockwise).
4. Sum the torques, taking into account their directions (positive for counterclockwise, negative for clockwise).

The net torque is the algebraic sum of all individual torques. This helps in understanding the overall rotational effect on the pulley.

Why does the weight of the pulley not contribute to the net torque?

The weight of the pulley does not contribute to the net torque because it acts through the axis of rotation. Torque is given by $\tau =Fr\sin \left(\theta \right)$, and when the force (weight) acts at the axis, the distance $r$ is zero. Therefore, the torque produced is zero. Additionally, forces that act along the axis of rotation do not create a lever arm, resulting in no rotational effect.

How do you calculate the torque produced by a force acting at the edge of a disc?

To calculate the torque produced by a force acting at the edge of a disc, use the formula:

$\tau =Fr\sin \left(\theta \right)$

where:

• $F$ is the force applied.
• $r$ is the radius of the disc (since the force is applied at the edge).
• $\theta$ is the angle between the force vector and the radius (usually 90 degrees, making $\sin \left(90\right)=1$).

This simplifies to $\tau =Fr$ when the force is perpendicular to the radius.

What factors affect the magnitude of torque on a disc or pulley?

The magnitude of torque on a disc or pulley is affected by three main factors:

• Force (F): The greater the force applied, the larger the torque.
• Distance (r): The distance from the axis of rotation to the point where the force is applied. A larger distance results in a greater torque.
• Angle (θ): The angle between the force vector and the lever arm. Torque is maximized when the angle is 90 degrees, as $\sin \left(90\right)=1$.

These factors combine in the formula $\tau =Fr\sin \left(\theta \right)$ to determine the overall torque.