Rotational Kinematics and Dynamics

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Theodore Bryan
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1 Rotational Kinematics and Dynamics Name : Date : Level : Physics I Teacher : Kim Angular Displacement, Velocity, and Acceleration Review - A rigid object rotating about a fixed axis through O perpendicular to the plane. - An arbitrary point P is at a fixed distance r from the origin and rotates about it in a circle or radius r. -The relationship between the arc length s, radius r and angle θ is θ = - θ is the ratio of an arc length and the radius of the circle with the unit radian, where one radian is the angle subtended by an arc length equal to the radius of the arc 1rad = 57.3 = 0.159revs - Angular speed is defined as w = and angular acceleration is defined as α = When rotating about a fixed axis, every particle on a rigid object rotates through the same angle and has the same angular speed and the same angular acceleration Rotational Kinematics: Rotational Motion with Constant Acceleration When an object s rotational motion is under constant angular acceleration, the kinematic relationships can be expressed as wf =wi + αt, θf = θi + wi t + αt², wf ²= wi² + 2α( θf θi ) *~Compare with kinematics : v f =v i + at, x f = x i + v i t + at², v f ²= v i ² + 2a(x f x i )

2 Q1) Rotating Wheel A wheel rotates with a constant angular acceleration of 3.5rad/s 2. If the angular speed of the wheel is 2rad/s at t i =0, (a) through what angle does the wheel rotate in 2secs? How many rotations does the wheel make? (b) What is the angular speed at t=2s? Ans) (a) 630, 1.75revs (b) 9rad/s Q2) A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of 12rad/s in 3s. (a) Find the angular acceleration of the wheel and (b) the angle (in radians) through which it rotates in this time. (c) How many rotations(=rev) does the wheel make through this time? Ans) 4rad/s 2, 18rads, 2.86revs Q3) An electric motor rotating a grinding wheel at 100rev/min is switched off. Assuming constant negative angular acceleration of 2rad/s 2, (a) how long does it take the wheel to stop? (b) Through how many rotation does it turn during the time found in part(a)? Ans) (a) 5.24s, 4.38revs

3 Angular and Linear Quantities - Point P moves in a circle where the linear velocity v is always tangent to the circular path. Hence, called tangential velocity. - The tangential speed and angular speed has a relationship as v = rw - The tangential acceleration and angular acceleration has a relationship as at = = r = rα => a t = rα - Tangential acceleration will only exist if point P is moving with changing tangential speed - Furthermore, if an object is moving in circular motion, there is another acceleration, called radial acceleration(or centripetal acceleration) ar = = rw² - The total acceleration will then be the combination of tangential acceleration and radial acceleration - Hence, the total acceleration for point P is a = = = Q4) A racing car travels on a circular track with a radius of 250m. If the car moves with a constant linear speed of 45m/s, find (a) its angular speed and (b) the acceleration. Ans) 0.18rad/s, 8.1m/s 2

4 Q5) A wheel 2m in diameter lies in vertical plane and rotates with a constant angular acceleration of 4rad/s 2. The wheel starts at rest at t=0, and the point P on the rim makes an angle of 57.3 with the horizontal at this time. At t=2s, find the (a) the angular speed of the wheel, (b) the linear speed and the acceleration of the point P, and (c) the angular position of the point P. Ans) 8rad/s, (b) 8m/s, a=64.12m/s 2 (c) 9rads Rotational Energy - The rotational kinetic energy of a rotating rigid object is expressed as K R =, where I = mi ri² : rotational inertia(=moment of inertia) *~Compare with K = - The moment of inertia(or rotational inertia) is a measure of the resistance of an object to changes in its rotational motion, just as mass is a measure of the resistance of an object to changes in its linear motion. - Note that mass is an intrinsic property of an object, where I depends on the physical arrangement of that mass Calculation of Rotational of Inertia I (revisit) I = mi ri² : moment of inertia - The moment of inertia for a point particle of mass m is expressed as I = mr² (Figure1) Figure1 - The moment of inertia for a point particle of mass m is expressed as I = 2mr² (Figure2) Figure2

5 Q7) Four Rotating Mass Four tiny spheres are fastened to the corners of a frame of negligible mass lying in the xy plane. We shall assume that the sphere s radii are small compared with the dimensions of the frame. The mass of M=2kg, m=1kg. (a) If the system rotates about the y-axis with an angular speed w=3rads, find the rotational of inertia and the rotational kinetic energy about this axis. (b) Suppose the system rotates in the xy plane about an axis through O(z-axis) with the same angular speed. Calculate the rotational of inertia and the rotational kinetic energy about this axis. M y m 3 O m M x Ans) (a) I=64kg m 2, K R =288J (b) I=82kg m 2, K R =369J Torque Revisit F = F 1 + F 2 + F N = ma τ = τ 1 + τ 2 + τ N = Iα 1. A meter-stick is place on a fulcrum at the 50cm mark. m 2 > m 1 10cm 50cm 90cm m 1 4kg m 2 τ = τ L + τ R = Iα => τ = r L F gl + r R F gr = Iα

6 Q8) Atwood s machine revisit. The pulley is free to rotate on a horizontal axis through its center. There is no slippage between the cord and the pulley. Assume the rotational inertial of the pulley is I = 8kg m 2 and the radius R=2m. The mass of m 1 is 2kg and m 2 is 4kg. Find the acceleration of both blocks and the tension in each side of the cord. m 1 R=2m T 2 T 1 m 2 Ans) 2.45m/s 2, T 1 =29.4N, T 2 =24.5N Q9) Two block system revisit. The frictional force between the block and table is 20N. The mass m 1 is 5kg and m 2 is 2.5kg. If the rotational inertia of the wheel is 4kg m 2 and the radius r=0.3m, find the acceleration of the blocks. Assume no slippage between the rope and wheel. *~notice that T 1 > T 2. This causes the pulley turn clockwise~* m 1 T 2 r T 1 m 2 Ans) 0.087m/s 2

7 Q10) A flywheel is a solid disk that rotates about an axis that is perpendicular to the disk at its center. Rotating flywheels provide means for storing energy in the form of rotational kinetic energy and are being considered as a possible alternative to batteries in electric cars. The gasoline burned in a 300-mile trip in a midsize car produces about J of energy. How fast would a 13kg flywheel with a radius of 0.3m have to rotate to store this much energy? Answer in rev/s Ans: rev/s Q11) Rotating Rod A uniform rod of length L=4m and mass M=1.5kg is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical plane. The rod is released from rest in the horizontal position. (a) What is the initial angular acceleration of the rod and initial linear acceleration of its right end? (b) What is the angular speed when the rod reaches its lowest position? (c) Determine the linear speed of the center of mass and the linear speed of the end of the rod when it reaches its lower position Ans) (a) α=3.68rad/s 2, a t =14.7m/s 2 (b) 2.71rad/s (c) v cm = 5.42m/s, v end =10.84m/s

8 The rotational of inertia for homogenous rigid bodies are expressed as below

9 Formula Summary If a particle rotates in a circle of radius r through an angle of θ(measured in radians), the arc length it moves through is s= rθ. s= r θ or θ = s / r, 1rad = 57.3 = 0.159revs Angular speed is defined as w = and angular acceleration is defined as α = When an object s rotational motion is under constant angular acceleration, the kinematic relationships can be expressed as w f =w i + αt, θ f = θ i + w i t + αt², w f ²= w i ² + 2α( θ f θ i ) When a rigid object rotates about a fixed axis, the angular position, angular speed, and angular acceleration are related to the linear position, linear speed, and linear acceleration through the relationships s= r θ, v = r w, a t = rα The rotational inertia of a system of particles I = i i 2 The rotational kinetic energy of a rotating rigid object with an angular speed of w can be expressed as K R = If a rigid object free to rotate about a fixed axis has a net external torque acting on it, the object undergoes an angular acceleration α, where τ = Iα Total linear acceleration will then the combination of tangential acceleration and radial acceleration a = a t + a r where a t is the tangential acceleration and a r is the centripetal(=radial) acceleration (a r = v 2 /r)

Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle