Intro to Torque: Videos & Practice Problems

Torque is the rotational equivalent of force, acting as a twist around an axis of rotation. It is calculated using the equation ${\tau }^{=}Frsin\theta$, where F is the applied force, r is the distance from the axis, and θ is the angle between the force and the radius vector. Maximum torque occurs when the force is applied perpendicularly and as far from the axis as possible, emphasizing the relationship between force, torque, and angular acceleration.

Intro to Torque

Torque is a fundamental concept in physics, representing the rotational equivalent of force. It can be understood as the twist that a force exerts on an object around a specific axis of rotation. For instance, when you push a door at its edge, it rotates around its hinges, which serve as the axis. The relationship between force, torque, and angular acceleration is crucial: a force can produce torque, which in turn can lead to angular acceleration.

To quantify torque, we use the equation:

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

In this equation, \(\tau\) represents torque, \(F\) is the applied force, \(r\) is the distance from the axis of rotation to the point where the force is applied, and \(\theta\) is the angle between the force vector and the position vector. The unit of torque is Newton-meters (N·m).

It is important to note that not all forces produce torque. For example, pushing directly at the hinge of a door (the axis of rotation) results in zero torque because the distance \(r\) is zero. Conversely, to maximize torque, one should apply force as far from the axis as possible and at a perpendicular angle (90 degrees) to the position vector. This configuration ensures that \(\sin(\theta)\) reaches its maximum value of 1, thereby maximizing torque.

When calculating torque, follow these steps: first, identify and draw the position vector \(r\) from the axis of rotation to the point of force application. Next, determine the angle \(\theta\) between the force vector and the position vector. Finally, substitute these values into the torque equation to find the torque produced.

For example, consider a door that is 3 meters wide with a force of 10 N applied in various ways. If the force is applied at the edge of the door (3 m from the hinge), the torque is:

\(\tau = 10 \, \text{N} \cdot 3 \, \text{m} \cdot \sin(90^\circ) = 30 \, \text{N·m}\)

However, if the force is applied halfway (1.5 m from the hinge), the torque is:

\(\tau = 10 \, \text{N} \cdot 1.5 \, \text{m} \cdot \sin(90^\circ) = 15 \, \text{N·m}\)

When the force is applied at the hinge, the torque is zero:

\(\tau = 10 \, \text{N} \cdot 0 \, \text{m} \cdot \sin(\theta) = 0 \, \text{N·m}\)

Understanding these principles of torque is essential for solving problems related to rotational motion and dynamics. Mastery of torque calculations will serve as a foundation for more advanced topics in physics.

You pull with a 100 N at the edge of a 25 cm long wrench, to tighten a bolt (gold), as shown. The angle shown is 53° . Calculate the torque your force produces on the wrench, about an axis perpendicular to it and through the bolt.

• A. 9.9 N•m
• B. 990 N•m
• C. 15 N•m
• D. 1500 N•m
• E. 20 N•m
• F. 2000 N•m

Answer: (C)

Here’s what students ask on this topic:

What is torque and how is it calculated?

Torque is the rotational equivalent of force, acting as a twist around an axis of rotation. It is calculated using the equation:

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

where $F$ is the applied force, $r$ is the distance from the axis, and $\theta$ is the angle between the force and the radius vector. Maximum torque occurs when the force is applied perpendicularly and as far from the axis as possible.

How does the angle of applied force affect torque?

The angle of the applied force significantly affects the torque produced. Torque is given by the equation:

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

where $\theta$ is the angle between the force and the radius vector. The sine function reaches its maximum value of 1 when $\theta$ is 90 degrees, meaning the force is applied perpendicularly to the radius vector. Therefore, to maximize torque, the force should be applied at a 90-degree angle to the radius vector.

Why is torque zero when force is applied at the axis of rotation?

Torque is zero when force is applied at the axis of rotation because the distance $r$ from the axis to the point of force application is zero. The torque equation is:

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

If $r$ is zero, then $\tau$ will also be zero regardless of the force $F$ or the angle $\theta$. This is because there is no lever arm to create rotational effect.

What are the units of torque?

The units of torque are Newton-meters (N·m). This is derived from the torque equation:

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

where force $F$ is measured in Newtons (N) and distance $r$ is measured in meters (m). The sine of the angle $\theta$ is dimensionless, so the resulting unit for torque is Newton-meters.

How can you maximize torque?

To maximize torque, you should apply the force as far from the axis of rotation as possible and at a 90-degree angle to the radius vector. The torque equation is:

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

Maximum torque occurs when $\theta$ is 90 degrees because $\sin \left(90\right)$ is 1. Additionally, increasing the distance $r$ from the axis increases the torque.