Rotational motion is the circular motion of an object about an axis of rotation. We will discuss specifically circular motion and spin. Circular motion is when an object moves in a circular path. Examples of circular motion include a race car speeding around a circular curve, a toy attached to a string swinging in a circle around your head, or the circular loop-the-loop on a roller coaster. Spin is rotation about an axis that goes through the center of mass of the object, such as Earth rotating on its axis, a wheel turning on its axle, the spin of a tornado on its path of destruction, or a figure skater spinning during a performance at the Olympics. Sometimes, objects will be spinning while in circular motion, like the Earth spinning on its axis while revolving around the Sun, but we will focus on these two motions separately. When solving problems involving rotational motion, we use variables that are similar to linear variables distance, velocity, acceleration, and force but take into account the curvature or rotation of the motion. Here, we define the angle of rotation, which is the angular equivalence of distance; and angular velocity, which is the angular equivalence of linear velocity. The pits dots along a line from the center to the edge all move through the same angle Δ θ Δ θ in time Δ t Δ t. The arc length, is the distance traveled along a circular path. The radius of curvature, r, is the radius of the circular path. The arc length, Δ s Δ s, is the distance covered along the circumference. In a given time, each pit used to record information on this line moves through the same angle. The angle of rotation is the amount of rotation and is the angular analog of distance. The angle of rotation Δ θ Δ θ is the arc length divided by the radius of curvature. Radians are actually dimensionless, because a radian is defined as the ratio of two distances, radius and arc length. A revolution is one complete rotation, where every point on the circle returns to its original position. One revolution covers 2 π 2 π radians or 360 degreesand therefore has an angle of rotation of 2 π 2 π radians, and an arc length that is the same as the circumference of the circle. See for the conversion of degrees to radians for some common angles. If an object rotates through a greater angle of rotation in a given time, it has a greater angular speed. The direction of the angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation. Angular velocity ω is the angular version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. Now, consider another example: the tire of a moving car see. Similarly, a larger-radius tire rotating at the same angular velocity, ω ω, will produce a greater linear tangential velocity, v, for unit for angular velocity car. This is because a larger radius means a longer arc length must contact the road, so the car must move farther in the same amount of time. The speed of the tread of the tire relative to the axle is v, the same as if the car were jacked up and the wheels spinning without touching the road. Because the road is stationary with respect to this point of the tire, the car must move forward at the linear velocity v. A larger angular velocity for the tire means a greater linear velocity for the car. However, there are cases where linear velocity and tangential velocity are not equivalent, such as a car spinning its tires on ice. In this case, the linear velocity will be less than the tangential velocity. Due to the lack of friction under the tires of a car on ice, the arc length through which the tire treads move is greater than the linear distance through which the car moves. Angular velocity ω and tangential velocity v are vectors, so we must include magnitude and direction. The direction of the angular velocity is along the axis of rotation, and points away from you for an object rotating clockwise, and toward you for an object rotating counterclockwise. In mathematics this is described by the right-hand rule. Tangential velocity is usually described as up, down, left, right, north, south, east, or west, as shown in. For an object traveling in a circular path at a constant speed, would the linear speed of the object change if the radius of the path increases. It is important that the circle be horizontal. Measure the time it takes in seconds for the object to travel 10 revolutions. Divide that time by 10 to get the angular speed in revolutions per second, which you can convert to radians per second. unit for angular velocity Describe what each graph looks like. If you swing an object slowly, it may rotate at less than one revolution per second. What would be the revolutions per second for an object that makes one revolution in five seconds. What would be its angular speed in radians per second. We can figure out the angle of rotation by multiplying a full revolution 2 π 2 π radians by the fraction of the 12 hours covered by the hour hand in going from 12 to 3. Discussion We were able to drop the radians from the final unit for angular velocity to part b because radians are actually dimensionless. This is because the radian is defined as the ratio of two distances radius and arc length. Thus, unit for angular velocity formula gives an answer in units of meters, as expected for an arc length. Discussion When we cancel units in the above calculation, we get 50. Because radians are dimensionless, we can insert them into the answer for the angular speed because we know that the motion is circular. Also note that, if an earth mover with much larger tires, say 1. They would have an angular speed of What is circular motion. Exercise 2 What is meant by radius of curvature when describing rotational motion. Identify three examples of an object in circular motion. What is the relative orientation of the radius and tangential velocity vectors of an object in uniform circular motion?.
