Rotational Motion Notes PDF

i. Rotational motion Able to define and write equations for the angular quantities in rotational motion. ii. Able to define and analyze rotational motion with constant angular acceleration problems. iii. Able to state and define the formula for...

i. Rotational motion Able to define and write equations for the angular quantities in rotational motion. ii. Able to define and analyze rotational motion with constant angular acceleration problems. iii. Able to state and define the formula for torque in terms of the angular acceleration and the moment of inertia. iv. Able to explain the effect of moment of inertia of an object undergoing rotational motion. Table below shows the symbols used in linear and rotational motion. Linear Rotational Quantity motion motion Displacement Initial velocity Final velocity Acceleration Time Angular Velocity, Acceleration, Rotational Kinematics In purely rotational motion, all points on the object move in circles around the axis (point O) of circle rotation. The radius of the circle is r. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined: where l= arc length (meter) and r = radius of the circle (meter) 3 One complete revolution has rotated through 3600 Angular displacement: (unit = rad= rev = 0) The average angular velocity is defined as the total angular displacement divided by time: (unit = rad/s= rev/s = rpm) The instantaneous angular velocity: The angular acceleration is the rate at which the angular velocity changes with time: (unit = rad/s2= rev/s2) The instantaneous acceleration: Every point on a rotating body has an angular velocity ω (rad/s) and a linear velocity or tangential v (m/s). They are related r is in meter Therefore, objects farther from the axis of rotation will move faster.  If the angular velocity of a rotating object changes, it has a tangential acceleration (linear acceleration): tangential acceleration : atan Even if the angular velocity is constant, each point on the object has a centripetal or radial acceleration : centripetal acceleration : aR The frequency is the number of complete revolutions per second. Frequencies are measured in hertz. The period is the time one revolution takes: Comparison between linear and rotational motion with constant acceleration. LINEAR MOTION ROTATIONAL MOTION Relation between linear & rotational motion a) The position : where in radians b) The speed : where : tangential / linear velocity ( in ms-1 ) : angular velocity ( in rads-1 ) c) The acceleration : where : tangential / linear acceleration ( in ms-2 ) : angular acceleration ( in rads-2 ) Example 1: A car is travelling with a velocity of 17.0 m s−1 on a straight horizontal highway. The wheels of the car has a radius of 48.0 cm. If the car then speeds up with an acceleration of 2.00 m s−2 for 5.00 s, calculate a. the number of revolutions of the wheels during this period, b. the angular speed of the wheels after 5.00 s. Solution : a. The initial angular velocity is and the angular acceleration of the wheels is given by Solution : a. By applying the equation of rotational motion with constant angular acceleration, thus therefore b. The angular speed of the wheels after 5.00 s is Example 2: The wheels of a bicycle make 30 revolutions as the bicycle reduces its speed uniformly from 50.0 km h-1 to 35.0 km h-1. The wheels have a diameter of 70 cm. a. Calculate the angular acceleration. b. If the bicycle continues to decelerate at this rate, determine the time taken for the bicycle to stop. Solution : Solution : a. The initial angular speed of the wheels is and the final angular speed of the wheels is therefore b. The car stops thus and Hence  Example 3: A blade of a ceiling fan has a radius of 0.400 m is rotating about a fixed axis with an initial angular velocity of 0.150 rev s-1. The angular acceleration of the blade is 0.750 rev s-2. Determine a. the angular velocity after 4.00 s, b. the number of revolutions for the blade turns in this time interval, c. the tangential speed of a point on the tip of the blade at time, t =4.00 s, d. the magnitude of the resultant acceleration of a point on the tip of the blade at t =4.00 s. Solution : a. Given t =4.00 s, thus Solution : b. The number of revolutions of the blade is c. The tangential speed of a point is given by Solution : d. The magnitude of the resultant acceleration is Example 4: A coin with a diameter of 2.40 cm is dropped on edge on a horizontal surface. The coin starts out with an initial angular speed of 18 rad s−1 and rolls in a straight line without slipping. If the rotation slows down with an angular acceleration of magnitude 1.90 rad s−2, calculate the distance travelled by the coin before coming to rest. Solution : The radius of the coin is Solution : The initial speed of the point at the edge the coin is and the final speed is The linear acceleration of the point at the edge the coin is given by Therefore the distance travelled by the coin is