Rotational Motion

This chapter deals with the concept of rotational motion as well as the kinematics, kinetic energy, torque, and angular energy involved. The focus in the chapter is primarily on rigid objects which have definite shapes and stay in fixed positions. The introduction builds on the idea of translational motion of an objects center of mass as covered in chapter 7. Rotational motion means that all the points in the given object are moving in circles, with the centers of the given circles they are moving in, lying in a line known as the axis of rotation as depicted in figure 8-1 a) and b). It is possible to calculate the angle of an arc through which the points travel using the equation ? = arc length (l)/distance between point P and the axis of rotation (r: radius of the circle). ?=l/r where the angle may be measured in rad, such that if l = r, ? = 1 rad which is approximately equal to 57.3o therefore 360o = 2?rad. Other concepts introduced in linear motion such as velocity and acceleration also apply in circular motion and are referred to as angular acceleration and angular velocity, remaining the same at each point within the rotating object.

The chapter then proceeds to define angular displacement as ?? = ?2 – ?1, which then means that angular velocity denoted as ? will be equal to angular displacement/change in time (? = ??/?t) given in radians per second, and is the same for each point in the object. Further, the convention is that displacement and velocity are positive when rotation is counterclockwise and vice versa. Angular acceleration denoted as ? is therefore calculated as ? = ??/?t.

The chapter then proceeds to relate velocity and linear distance covered using the equation

v = ?l/?t = and considering ?l = r??, then v = r??/?t =r?

Changes in angular velocity may result in linear acceleration which can be calculated depending on whether its direction is tangent to the given point’s circular path. Angular acceleration relates to tangential linear acceleration as follows atan=r?. The linear acceleration of the given point will therefore be the sum total of tangential and radial acceleration (centripetal acceleration) calculated as a=atan+aR and aR=?2r. This therefore means that the centripetal force becomes greater the farther from the axis a point is.

The next concept introduced is that of frequency, which refers to the number of complete revolutions achieved per second, given by the equation f = ?/2?, where 1Hz =1revolution per second. The time require for a complete revolution (period) is denoted as T, which is equal to 1/f.

Similar to linear acceleration the concept of constant angular acceleration also exists, calculated using the equation Constant Angular Acceleration = (? + ?0)/2 where ?0 is angular acceleration at t=0

The chapter introduces the concept of torque denoted as ?, which essentially refers to the force times lever arm required to start the rotational motion of an object, with figure 8-10 highlighting the concept of the lever arm and that the closer a force is applied to the axis, the harder it is to initiate motion and vice versa. Such that ? = r1F and ? =rFsin?, where ? is the angle between the direction of force and r. Relating rotational motion to linear motion essentially means that rotational kinetic energy can be obtained using the equation 1/2I?2 where I is inertia given by the equation I=?mr2.