Unit 6 takes the motion ideas you already know and applies them to circles and spinning objects. After it, you should be able to describe rotation with angles and angular velocity, find the inward force that keeps something moving in a circle, and use torque to predict how an object's spin changes.
Start with the overview for the big picture, then work through each topic's notes and answer the checkpoint question before you read its answer.
Wheels and fans spin around an axis, while cars on curves and satellites in circular orbits follow curved paths. This unit gives you the tools to describe that motion and explain what causes it. Most of the math is the kinematics and Newton's laws you already know, rewritten with angular variables.
Honors Physics Unit 6 covers circular and rotational motion. You measure rotation in radians, connect angles to arc length with , and connect angular velocity to linear speed with . You analyze uniform circular motion using and a net inward force of . The unit finishes with rotational kinematics, torque (), moment of inertia, and the rotational version of Newton's second law, . Order and depth vary by school, and some courses move torque to a later unit.
Displacement becomes angle , velocity becomes angular velocity , and acceleration becomes angular acceleration . The radius links each pair, as in and . The constant-acceleration equations carry over with the symbols swapped.
Velocity includes direction, and direction changes constantly on a circle. That change is centripetal acceleration, , pointing toward the center. Newton's second law then requires a net force pointing toward the center, supplied by real forces like tension, friction, or gravity.
The same push can open a door easily or barely move it, depending on where and at what angle you push. Torque, , captures that. How much a given torque speeds up a spin depends on the moment of inertia, through .
Once you see that , , and behave like , , and , the unit becomes a translation exercise. Torque acts like force and moment of inertia acts like mass, so mirrors . Circular motion also follows Newton's second law. Its acceleration has a component toward the center and can also have a tangential component when speed changes.
The angle of rotation \theta tells you how far something has turned, the way displacement tells you how far it has moved.
In uniform circular motion the speed stays constant but the direction keeps changing, so the velocity changes and the object accelerates.
Rotational kinematics works like linear kinematics with the symbols swapped.
The angle of rotation tells you how far something has turned, the way displacement tells you how far it has moved. Physics uses radians because they make the link to distance clean. One full turn is rad, or 360°, so 1 rad is about 57.3°.
A point at radius that turns through an angle of magnitude in one direction travels an arc length , with the angle in radians. Average angular velocity is , measured in rad/s. At any instant, the point's linear speed is , where is its instantaneous angular velocity.
Every point on a rigid spinning object has the same . Points farther from the axis still move faster, because they cover a bigger circle in the same time. For constant angular speed, the period is the time for one turn and the frequency is . The angular speed is then .
Worked example. A ceiling fan spins at 120 revolutions per minute, and its blade tips are 0.60 m from the axis. Convert to rad/s first: rad/s. The tip speed is m/s. The period is s, which matches 120 turns in 60 seconds.
Check yourself. Two kids ride a merry-go-round, one 1 m from the center and one 2 m from the center. Who has the larger angular velocity, and who has the larger speed? (Answer: They have the same angular velocity because the whole platform turns together, but the kid at 2 m has twice the linear speed since .)
In uniform circular motion the speed stays constant but the direction keeps changing, so the velocity changes and the object accelerates. The velocity always points along the tangent to the circle. The acceleration points toward the center and is called centripetal acceleration, with magnitude .
By Newton's second law, an inward acceleration needs an inward net force of size . That net inward force is what people call the centripetal force. It is not a new kind of force. Tension, friction, gravity, or the normal force supplies it, alone or in combination. If the net force becomes zero, the object continues in a straight line along the tangent.
Because depends on , doubling your speed on the same curve takes four times the inward force.
Worked example. A 1000 kg car rounds a flat curve of radius 50 m at 15 m/s. The required inward force is N, and static friction from the road supplies it. Friction can provide at most , so setting gives a minimum coefficient of . Mass cancels, so a heavier car needs the same to make this turn.
Check yourself. A 0.50 kg puck on a 0.80 m string moves in a horizontal circle on a frictionless table at 4.0 m/s. What is the tension in the string? (Answer: Tension is the only inward force, so N.)
| Situation | Real force pointing toward the center |
|---|---|
| Puck on a string on a frictionless table | Tension in the string |
| Car turning on a flat road | Static friction from the road on the tires |
| Satellite in circular orbit | Earth's gravitational pull |
| Car on a banked curve with no friction | Horizontal component of the normal force |
Rotational kinematics works like linear kinematics with the symbols swapped. For constant angular acceleration, use , , and . A point on a wheel that is speeding up has tangential acceleration , which changes its speed, plus centripetal acceleration , which changes its direction.
Torque is what changes rotation. Its size is , where is the distance from the axis to where the force acts and is the angle between and . Pushing perpendicular to a door at the handle gives the most torque, and pushing straight toward the hinge gives zero.
Moment of inertia is the rotational version of mass. For a point mass at distance from an axis, . For a solid disk or thin ring about an axis through its center and perpendicular to its plane, or , respectively. For rotation about a fixed axis, Newton's second law becomes .
Worked example. A string pulls with 10 N tangent to the rim of a 4.0 kg solid disk of radius 0.20 m that starts at rest. The disk rotates about a fixed, frictionless axle through its center, perpendicular to its face. The torque is N·m. The moment of inertia is kg·m², so rad/s². After 2.0 s, rad/s.
Check yourself. A solid disk and a thin ring have the same mass and radius. Each rotates about a fixed axis through its center and perpendicular to its plane, and the same net torque acts on each. Which spins up faster? (Answer: The solid disk. Its is half the ring's , so with it gets twice the angular acceleration.)
| Linear | Rotational | Connection |
|---|---|---|
| Displacement $x$ (m) | Angle $\theta$ (rad) | $s = r\theta$ |
| Velocity $v$ (m/s) | Angular velocity $\omega$ (rad/s) | $v = r\omega$ |
| Acceleration $a$ (m/s²) | Angular acceleration $\alpha$ (rad/s²) | $a_t = r\alpha$ |
| Force $F$ (N) | Torque $\tau$ (N·m) | $\tau = rF\sin\theta$ |
| Mass $m$ (kg) | Moment of inertia $I$ (kg·m²) | $I = \sum mr^2$ |
| $\sum F = ma$ | $\sum\tau = I\alpha$ | Same form, angular variables |
| Term | Definition |
|---|---|
| Radian | The unit of angle in physics, with $2\pi$ rad in one full revolution. |
| Angular velocity | The rate at which an object's angle changes, in rad/s. Average angular velocity is $\omega_{\text{avg}} = \Delta\theta/\Delta t$. |
| Period | The time for one complete revolution. At constant angular speed, $T = 2\pi/|\omega|$. |
| Centripetal acceleration | The acceleration toward the center of a circular path, with magnitude $v^2/r$. |
| Centripetal force | The net inward force, $mv^2/r$, that real forces must supply for an object to move in a circle. |
| Tangential acceleration | The component of acceleration along the path that changes an object's speed, equal to $r\alpha$. |
| Angular acceleration | The rate of change of angular velocity, in rad/s². Average angular acceleration is $\alpha_{\text{avg}} = \Delta\omega/\Delta t$. |
| Torque | The turning effect of a force about an axis, $\tau = rF\sin\theta$, in N·m. |
| Moment of inertia | An object's resistance to changes in rotation, set by its mass and how far that mass is from the axis. |
The name makes it sound like its own force, so it is easy to add it next to tension or friction and count it twice. Draw only real forces, then set the sum of their inward components equal to .
Riding in a turning car feels like being pushed outward, but that feeling is your body trying to keep going straight. A ball whose string snaps initially moves along the tangent to the circle, in the direction its velocity was already pointing. Its later path can curve if another force, such as gravity, acts on it.
These formulas only work with radians, because the radian is defined by arc length divided by radius. Convert first, using rad per revolution and 60 s per minute, so 30 rpm becomes rad/s.
Torque is not simply unless the force is perpendicular to the line from the axis. Include , or use only the force component perpendicular to , and a force aimed straight at the axis gives zero torque.