ROTATIONAL MOTIONUntil now we have been looking at translational motion, motion in which something shifts in positionfrom one moment to the next. But rotational motion is just as common. The wheels, pulleys, propellers,drills, and audio disks rotate in order to do their job. Here our concern is mainly with the rotationalmotion of rigid body, one whose shape does not change as its spins.Angular MeasureWe are accustomed to measuring angles in degrees of a full rotation. That is, a complete turn represents360o. A better unit for our present purposes is the radian (rad). The radian is defined with the help of acircle drawn with its center at the vertex of the angle in question.Angular Speed, ωIf a rotating body turns through the angle θ in the time t, its average angular speed ω, is:
Angular Speed and Linear SpeedSuppose we have a particle moving with the uniform speed v in a circle of radius r. The particle travelsthe distance s=vt in time t. The angular distance through which it moves in that time is:Angular Acceleration, αA rotating body need not to have a uniform angular speed ω. The angular speed of a body changes by anamount Δω in the time interval Δt, its average angular acceleration α, is:Angular Acceleration and Linear AccelerationThe acceleration of a particle can be expressed in terms of its normal and tangential components.
We must be careful to distinguish between tangential acceleration at of a particle, which represents achange in speed, and its normal acceleration an (also known as centripetal acceleration), whichrepresents the change in its direction of motion. This normal acceleration is directed toward the centerof its circular path.Example 1:A particle is moving in a circle of radius 0.40 m at the instant when the angular speed is 2.0 rad/s and theangular acceleration is 5.0 rad/s2, find: (a) The linear speed. (b) The magnitude of the total acceleration.
COMPARISON WITH LINEAR MOTIONThe formulas we obtained for linear motion of a particle under constant acceleration all havecounterparts in angular motion. Because the derivation are the same, they are simply listed in tablebelow,Example 2:A motor starts rotating from rest with an angular acceleration of 12.0 rad/s2. (a) What is the motor’sangular speed 4.0s later, in radians per second? (b) What is the motor’s angular speed at this period, inrevolutions per minute? (c) How many revolution does it make in this period of time?
Example 3:A race car C travels around the horizontal track that has a radius of 90m. If the car increases its speed ata constant rate of 2 m/s2, starting from rest, determine the time needed for it to reach an accelerationof 3 m/s2, in seconds. What is its speed at this instant, in m/s?Problems: 1. A car makes a U-turn in 5.0s. What is its average angular speed? 2. The shaft of a motor rotates at a constant angular speed of 3000 rpm. How many radians will it have turned through in 10s? 3. The blades of a rotary lawnmower are 30 cm long and rotate at 315 rad/s. Find the linear speed of the blade tips and their angular speed in rpm. 4. A drill bit 0.25-in in diameter is turning at 1200 rpm. Find the linear speed of a point on its circumference in ft/sec. 5. A steel cylinder 40 mm in radius is to be machined in a lathe. At how many revolutions per second should it rotate in order that the linear speed of the cylinder’s surface be 70 cm/s?