# Circular Motion & Rotation

Up to this point we have dealt solely with the physics involved in translational motion, the movement of an object from one position to another. In this lesson, we will first cover a special type of translational motion called circular motion. Then we will delve into the physics involved in rotational motion, the turning of an object around a point.

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### Uniform Circular Motion

- Circular motion occurs when an object follows a circular path, such as when you swing a yo-yo around in circles.
- Uniform circular motion is a special type of circular motion in which the object's speed along that path is constant.
- Note that, although an object in uniform circular motion has a constant speed, it does not have a constant velocity, because the direction of motion is changing constantly as it moves around the circle.
- Because the object's velocity is changing, the object must experience an acceleration, which acts only to change the direction of the velocity, not its magnitude. Since there is an acceleration acting on the object, by Newton's First Law, there must be a force acting on the object.
- Diagram
**a.**below shows an object in uniform circular motion, with its velocity**v**shown at time_{1}*t*_{1}, and its velocity**v**shown a short period of time later,_{2}*t*_{2}. Notice that the lengths of both velocity vectors are the same, indicating that their magnitudes are equal. **a.****b.**- We know from kinematics that an object's acceleration can be calculated using the formula
**a**=**Δv**/*Δt*. To find**Δv**in this situation, you would subtract**v**from_{1}**v**. The vector representation of this subtraction is shown in diagram_{2}**b.**above. - Notice that the vector representing
**Δv**points toward the center of the circular path. Remember that, since time has no direction, the direction of acceleration is the same as the direction of velocity, so the acceleration of an object in uniform circular motion is directed toward the center of the circular path. - This acceleration, which keeps the object moving in a circle, is called
*centripetal acceleration*. Centripetal acceleration can be found using the following equation:*a*_{c}=*v*_{t}^{2}/*r**v*is the_{t}*tangential speed*of the object. Tangential speed simply means the magnitude of the object's velocity while the direction is tangent to the circular path, which is true as long as the object is in uniform circular motion.*r*is the radius of the circular path.

- We can plug this equation into Newton's Second Law to find the magnitude of the force causing this acceleration, which is called the
*centripetal force*:*F*_{c}=*ma*_{c}=*mv*_{t}^{2}/*r*- Note that the direction of force is always the same as the direction of the acceleration it causes. So, centripetal force acts toward the center of the circular path.

- For an object tied to a string and spun in a circle at constant speed, there is a maximum tangential speed at which the object can travel before the string breaks. If the force required to break the string is known, the maximum speed of the object can be found by solving the above equation for
*v*._{t} - If the string attached to this object breaks, the object will continue in the direction it was moving at the instant the string snapped, and immediately begin to exibit projectile motion.
- An object does not have to be attached to a string to experience circular motion and centripetal force. For example, a car traveling at constant speed along a circular track is in uniform circular motion, and the centripetal force is provided by friction.
- For an object in uniform circular motion whose circular path is vertical, the force of gravity must also be taken into account. Also note that for such an object that is not attached to a string, but instead moving along the inside of a vertical, circular track (such as a roller coaster in a loop), the centripetal force on the object is provided by the normal force exerted on the object by the track.

### Rotational Kinematics

- In the above diagram, the circle is rotated as shown by the arrows, so that each point along the horizontal axis ends on the corresponding point on the vertical axis.
- Since each point along the surface of the disc (assuming the circle to be a physical object with some third dimension) moves through the same angle, the disc is considered a rigid body. A rigid body is defined as an object such that each point along a given radius undergoes the same angular displacement when rotated.
- Angular displacement, denoted Δα, is the angle through which a point moves when rotated about some other point. Note that, unlike other angles in physics, angular displacement is measured in radians.
- Angles can be converted to radians by multiplying the angle measure, in degrees, times π/180. So, in the diagram above, the angular displacement of each point from start to finish is 90
^{o}= 90π/180 = π/2 radians. - Angular displacement is related to the arc length of the rotation (the distance traveled by a given point), denoted
*Δs*, by the equation Δα =*Δs*/*r*. - Remember that the rate of change of displacement with respect to time is velocity. Similarly, the rate of change of angular displacement with respect to time is angular velocity, denoted by the lowercase Greek letter omega (ω).
- Average angular velocity can be found using the following equation: ω
_{avg}= Δα/*Δt*. The instantaneous angular velocity can be found by taking the derivative with respect to time of angular displacement, ω =*d*α/*dt*. - Since we know that all points on a rigid body undergo the same angular displacement, we can also infer that for any given rotation, all points on a rigid body will have the same angular velocity, ω. However, they do not have the same linear velocity,
*v*.

### Torque

- Torque is, put simply, how effective a force is at causing a rotational acceleration. For example, consider an unlocked door which is opened simply by pushing on the door to rotate it about its hinges. If you push on the door as far from its hinges as possible, the door opens easily with relatively little force. However, if you push close to the hinges, it requires a lot more force to cause the same rotation of the door. So, pushing further from the hinges produces a greater torque on the door.
- Any rotating object has a pivot, which is the point around which the object rotates. As an example, the pivot on a door is its hinges, and the pivot of your arm is your shoulder.
- As you can see in the door example, the torque produced by a force acting on a rotating object increases as the distance from the object's pivot increases. You also know from experience that pushing harder on a door will cause it to open faster, so we can see that torque also increases as the magnitude of the force causing the rotation increases.
- From these facts, we can determine that the basic definition of torque, represented by the Greek letter tau (
*t*), can be represented by the equation*t*=*Fr*, where*F*is the magnitude of the force applied and*r*is the radius between the object's pivot and the point at which the force is acting on the object. - Though this definition works when the force applied is perpendicular to the radius, a force that is not perpendicular to the radius (such as pushing on a door from an angle instead of straight-on) must be resolved into its components, and only the component perpendicular to the radius is used. For an acute angle θ measured between
**F**and**r**, this equation takes the form*t*=*rF*sinθ. - Torque can be produced by a force that causes an object to rotate regardless of whether it is direct, such as pushing on a door, or indirect, such as pulling a rope to open the door.
- Remember that forces which produce torque cause the object they act on to rotate, so two forces that act on the same object but on opposite sides of the pivot may not be as straight forward as translational motion. If these two forces both act upward on the object, they actually produce torque in opposite directions: one clockwise, one counter-clockwise. If one points up and one points down, they actually both produce torque in the same direction, either clockwise or counter-clockwise.
- The individual torques produced by multiple forces acting on the same object can be added together, as long as you keep in mind their clockwise or counter-clockwise directions. Remember that those forces which are not perpendicular to their respective radii must be resolved into components before their torques can be found.

### Equilibrium

- In the lesson on forces, we discussed the concept of equilibrium, specifically translational equilibrium. Translational equilibrium occurs when the sum of all forces in the x direction and the sum of all forces in the y direction are both zero.
- Similarly, we can discuss rotational equilibrium as the state in which the sum of all torques acting on a system is equal to zero.
- When calculating the sum of torques, keep in mind that torques in the counter-clockwise direction are considered positive, and torques in the clockwise direction are considered negative.
- If an object is in both translational and rotational equilibrium, then both need to be properly taken into account. Translational equilibrium should be calculated without respect to the direction of rotation of torques, while rotational equilibrium should be calculated with respect to the direction of rotation of torques.