Rotation, Torque, and Balance

A small force can turn a stubborn bolt when it acts far enough from the pivot. Rotation depends on force, distance, and the angle at which the force is applied.

This lesson supports Chapter 8 of Physics for Beginners. Keep the physical picture in view, write the units, and check the direction of every vector before trusting a calculation.

Angular quantities describe rotation

Angular displacement is measured in radians. Angular velocity tells how quickly the angle changes, and angular acceleration tells how quickly angular velocity changes. For a point a distance \(r\) from the axis, tangential speed is \(v=r\omega\). Points farther from the axis move faster even though the whole object shares one angular velocity.

Torque changes rotational motion

The magnitude of torque is \[\tau=rF\sin\theta.\] The lever arm is the perpendicular distance from the axis to the force’s line of action. A force aimed directly through the pivot produces no torque. The rotational form of Newton’s second law is \(\tau_{\text{net}}=I\alpha\).

Balance requires two conditions

An object in static equilibrium has zero net force and zero net torque. Checking only the forces is not enough. An object resists tipping while the vertical line through its center of mass falls inside its support base. A wider base and a lower center of mass increase the tilt needed to move that line beyond the base.

Worked example

A \(20\text{ N}\) force acts perpendicular to a \(0.30\text{ m}\) wrench. The torque magnitude is \(\tau=rF=0.30(20)=6.0\text{ N}\cdot\text{m}\). The same force applied halfway down the wrench would produce half as much torque.

Watch the idea in action

The Organic Chemistry Tutor gives a focused explanation of the chapter’s central idea. Pause before each calculation and predict the next step on paper.

Practice questions

1. Write the torque magnitude formula.
2. When is torque largest for fixed \(r\) and \(F\)?
3. What is the relation between tangential speed and angular velocity?
4. What two conditions define static equilibrium?
5. Why does a longer wrench help loosen a bolt?
6. What happens when a force’s line of action passes through the pivot?

Answers

1. \(\tau=rF\sin\theta\).
2. When the force is perpendicular to the radius.
3. \(v=r\omega\).
4. Zero net force and zero net torque.
5. It increases the lever arm and torque.
6. The torque is zero.

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