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DF015 CHAPTER 8 CHAPTER 8: Rotation of a rigid body (8 Hours) 1
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DF025 CHAPTER 8 Learning Outcome: 8.1 Rotational kinematics (2 hours) At the end of this chapter, students should be able to: a) Define and use: angular displacement () average angular velocity (av) instantaneous angular velocity () average angular acceleration (av) instantaneous angular acceleration (). 2
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DF025 CHAPTER 8 b) Relate parameters in rotational motion with their corresponding quantities in linear motion. Write and use; v2 s rθ ; v r; at r ; ac r 2 r c) Use equations for rotational motion with constant angular acceleration; 1 , θ ω t t 2, ω2 ω 2 2αθ . ω ω αt 0 0 0 2 3
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DF025 CHAPTER 8 8.1 Rotational kinematics a)(i) Angular displacement, is defined as an angle through which a point or line has been rotated in a specified direction about a specified axis. The S.I. unit of the angular displacement is radian (rad). Figure 7.1 shows a point P on a rotating compact disc (CD) moves through an arc length s on a circular path of radius r about a fixed axis through point O. Figure 7.1 4
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DF025 CHAPTER 8 From Figure 7.1, thus s θ OR s rθ r where θ : angle (angular displaceme nt) in radian s : arc length r : radiusof the circle Others unit for angular displacement is degree () and revolution (rev). Conversion factor : 1 rev 2π rad 360 Sign convention of angular displacement : Positive – if the rotational motion is anticlockwise. Negative – if the rotational motion is clockwise. 5
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DF025 CHAPTER 8 a)(ii)(iii) Angular velocity Average angular velocity, av is defined as the rate of change of angular displacement. Equation : θ2 θ1 θ ωav t 2 t1 t whereθ2 : final angular displaceme nt in radian θ1 : initial angular displaceme nt in radian t : time interval Instantaneous angular velocity, is defined as the instantaneous rate of change of angular displacement. Equation : θ dθ limit t 0 t dt 6
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DF025 CHAPTER 8 It is a vector quantity. The unit of angular velocity is radian per second (rad s-1) Others unit is revolution per minute (rev min1 or rpm) Conversion factor: 2 1 1 rpm rad s rad s 1 60 30 Note : Every part of a rotating rigid body has the same angular velocity. Direction of the angular velocity Its direction can be determine by using right hand grip rule where Thumb : direction of angular velocity Curl fingers : direction of rotation 7
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DF025 CHAPTER 8 Figures 7.2 and 7.3 show the right hand grip rule for determining the direction of the angular velocity. Figure 7.2 Figure 7.3 8
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DF025 CHAPTER 8 Example 1 : The angular displacement, of the wheel is given by θ 5t 2 t where in radians and t in seconds. The diameter of the wheel is 0.56 m. Determine a. the angle, in degree, at time 2.2 s and 4.8 s, b. the distance that a particle on the rim moves during that time interval, c. the average angular velocity, in rad s1 and in rev min1 (rpm), between 2.2 s and 4.8 s, d. the instantaneous angular velocity at time 3.0 s. 9
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DF025 CHAPTER 8 d 0.56 Solution : r 0.28 m 2 2 a. At time, t1 =2.2 s : θ1 52.2 2.2 2 θ1 22 rad 180 θ1 22 rad 1261 π rad At time, t2 =4.8 s : θ2 54.8 4.8 2 θ2 110 rad 180 θ2 110 rad 6303 π rad 10
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DF025 CHAPTER 8 d 0.56 Solution : r 0.28 m 2 2 b. By applying the equation of arc length, s rθ Therefore s r r 2 1 s 0.28110 22 s 24.6 m c. The average angular velocity in rad s1 is given by θ 2 1 ωav t t2 t1 ωav 110 22 4.8 2.2 ωav 33.9 rad s 1 11
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DF025 CHAPTER 8 Solution : c. and the average angular velocity in rev min1 is 33.9 rad 1 rev 60 s ωav 1 s 2 rad 1 min ωav 324 rev min 1 OR 324 rpm d. The instantaneous angular velocity as a function of time is dθ ω dt ω dt d 2 5t t ω 10t 1 ω 103.0 1 At time, t =3.0 s : ω 29 rad s 1 12
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DF025 CHAPTER 8 Example 2 : A diver makes 2.5 revolutions on the way down from a 10 m high platform to the water. Assuming zero initial vertical velocity, calculate the diver’s average angular (rotational) velocity during a dive. (Given g = 9.81 m s2) Solution : uy 0 θ0 0 10 m θ1 2.5 rev water 13
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DF025 CHAPTER 8 Solution : θ1 2.5 2π 5π rad From the diagram, s y 10 m Thus 1 s y u y t gt 2 2 10 0 9.81t 2 1 2 t 1.43 s Therefore the diver’s average angular velocity is θ1 θ0 ωav t 5π 0 ωav 1.43 ωav 11.0 rad s 1 14
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DF025 CHAPTER 8 a)(iv)(v) Angular acceleration Average angular acceleration, av is defined as the rate of change of angular velocity. Equation : ω2 ω1 ω av t 2 t1 t where ω2 : final angular velocity ω1 : initial angular velocity t : time interval Instantaneous angular acceleration, is defined as the instantaneous rate of change of angular velocity. Equation : ω dω α limit t 0 t dt 15
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DF025 CHAPTER 8 It is a vector quantity. The unit of angular acceleration is rad s2. Note: If the angular acceleration, is positive, then the angular velocity, is increasing. If the angular acceleration, is negative, then the angular velocity, is decreasing. Direction of the angular acceleration If the rotation is speeding up, and in the same direction as shown in Figure 7.4. α Figure 7.4 16
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DF025 CHAPTER 8 If the rotation is slowing down, and have the opposite direction as shown in Figure 7.5. α Figure 7.5 Example 3 : The instantaneous angular velocity, of the flywheel is given by ω 8t 3 t 2 where in radian per second and t in seconds. Determine a. the average angular acceleration between 2.2 s and 4.8 s, b. the instantaneous angular acceleration at time, 3.0 s. 17
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DF025 CHAPTER 8 Solution : a. At time, t1 =2.2 s : ω1 82.2 2.2 3 2 ω1 80.3 rad s 1 At time, t2 =4.8 s : ω2 84.8 4.8 3 2 ω2 862 rad s 1 Therefore the average angular acceleration is ω2 ω1 αav t 2 t1 862 80.3 αav 4.8 2.2 αav 301 rad s 2 18
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DF025 CHAPTER 8 Solution : b. The instantaneous angular acceleration as a function of time is dω α dt α dt d 3 2 8t t α 24t 2 2t At time, t =3.0 s : α 243.0 23.0 2 α 210 rad s 2 19
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DF025 CHAPTER 8 Exercise 8.1a : 1. If a disc 30 cm in diameter rolls 65 m along a straight line without slipping, calculate a. the number of revolutions would it makes in the process, b. the angular displacement would be through by a speck of gum on its rim. ANS. : 69 rev; 138 rad 2. During a certain period of time, the angular displacement of a swinging door is described by θ 5.00 10.0t 2.00t 2 where is in radians and t is in seconds. Determine the angular displacement, angular speed and angular acceleration a. at time, t =0, b. at time, t =3.00 s. ANS. : 5.00 rad, 10.0 rad s1, 4.00 rad s2; 53.0 rad, 22.0 rad s1, 4.00 rad s2 20
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DF025 CHAPTER 8 Learning Outcome: 8.1.b) Relationship between linear and rotational motion (½ hour) At the end of this chapter, students should be able to: Relate parameters in rotational motion with their corresponding quantities in linear motion. Write and use; v2 s rθ ; v r; at r ; ac r 2 r 21
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DF025 CHAPTER 8 8.1.b) Relationship between linear and rotational motion Relationship between linear velocity, v and angular velocity, When a rigid body is rotates about rotation axis O , every particle in the body moves in a circle as shown in the Figure 7.6. y v P r s x O Figure 8.6 22
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DF025 CHAPTER 8 Point P moves in a circle of radius r with the tangential velocity v where its magnitude is given by ds v and s rθ dt d vr dt v r The direction of the linear (tangential) velocity always tangent to the circular path. Every particle on the rigid body has the same angular speed (magnitude of angular velocity) but the tangential speed is not the same because the radius of the circle, r is changing depend on the position of the particle. Simulation 8.1 23
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DF025 CHAPTER 8 Relationship between tangential acceleration, at and angular acceleration, If the rigid body is gaining the angular speed then the tangential velocity of a particle also increasing thus two component of acceleration are occurred as shown in Figure 7.7. y at P a ac O x Figure 8.7 24
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DF025 CHAPTER 8 The components are tangential acceleration, at and centripetal acceleration, ac given by dv at and v rω dt d at r at r dt v2 but ac r v2 r The vector sum of centripetal and tangential acceleration of a particle in a rotating body is resultant (linear) acceleration, a given by a at ac Vector form and its magnitude, a at ac 2 2 25
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DF025 CHAPTER 8 Learning Outcome: 8.1.c) Rotational motion with uniform angular acceleration (1/2 hour) At the end of this chapter, students should be able to: Write and use equations for rotational motion with constant angular acceleration; ω ω0 αt 1 2 θ ω0 t t 2 ω2 ω0 2αθ 2 26
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DF025 CHAPTER 8 8.1.c) Rotational motion with uniform angular acceleration Table 8.1 shows the symbols used in linear and rotational kinematics. Linear Rotational Quantity motion motion s Displacement θ u Initial velocity ω0 v Final velocity ω a Acceleration α t Time t Table 8.1 27
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DF025 CHAPTER 8 Table 8.2 shows the comparison of linear and rotational motion with constant acceleration. Linear motion Rotational motion a constant α constant v u at ω ω0 αt 1 2 1 2 s ut at θ ω0 t αt 2 2 v 2 u 2 2as ω 2 ω0 2 2αθ s v u t 1 θ ω ω0 t 1 2 2 where in radian. Table 8.2 28
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DF025 CHAPTER 8 Example 4 : A car is travelling with a velocity of 17.0 m s1 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 s2 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 : u 17.0 m s 1 , r 0.48 m, a 2.00 m s 2 , t 5.00 s a. The initial angular velocity is u rω0 1 17.0 0.48ω0 ω0 35.4 rad s and the angular acceleration of the wheels is given by a rα 2.00 0.48α α 4.17 rad s 2 29
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DF025 CHAPTER 8 Solution : u 17.0 m s 1 , r 0.48 m, a 2.00 m s 2 , t 5.00 s a. By applying the equation of rotational motion with constant angular acceleration, thus 1 2 θ ω0 t αt 2 θ 35.4 5.00 4.17 5.00 1 2 2 θ 229 rad therefore 1 rev θ 229 rad 36.5 rev 2π rad b. The angular speed of the wheels after 5.00 s is ω ω0 αt ω 35.4 4.17 5.00 ω 56.3 rad s 1 30
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DF025 CHAPTER 8 Example 5 : 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. 0.70 Solution : θ 30 2π 60π rad, r 0.35 m, 2 50.0 km 10 3 m 1 h u 1 13.9 m s , 1 h 1 km 3600 s 35.0 km 10 3 m 1 h v 9.72 m s 1 1 h 1 km 3600 s 31
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DF025 CHAPTER 8 Solution : a. The initial angular speed of the wheels is u rω0 13.9 0.35ω0 ω0 39.7 rad s 1 and the final angular speed of the wheels is v rω 9.72 0.35ω ω 27.8 rad s 1 therefore ω2 ω0 2αθ 2 27.8 39.7 2α60π 2 2 α 2.13 rad s 2 1 b. The car stops thus ω 0 and ω0 27.8 rad s Hence ω ω0 αt 0 27.8 2.13t t 13.1 s 32
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DF025 CHAPTER 8 Example 6 : 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 : r 0.400 m, ω0 0.150 2π 0.300π rad s 1 , α 0.750 2π 1.50π rad s 2 a. Given t =4.00 s, thus ω ω0 αt ω 0.300π 1.50π 4.00 1 ω 19.8 rad s 33
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DF025 CHAPTER 8 Solution : b. The number of revolutions of the blade is 1 2 θ ω0 t αt 2 θ 0.300 4.00 1.504.00 1 2 2 θ 41.5 rad 1 rev θ 41.5 rad 6.61 rev 2π rad c. The tangential speed of a point is given by v rω v 0.400 19.8 v 7.92 m s 1 34
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DF025 CHAPTER 8 Solution : d. The magnitude of the resultant acceleration is a ac at 2 2 2 v 2 a rα 2 r 7.92 2 2 a 0.400 1.50π 2 0.400 a 157 m s 2 35
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DF025 CHAPTER 8 Example 7 : Calculate the angular velocity of a. the second-hand, b. the minute-hand and c. the hour-hand, of a clock. State in rad s-1. d. What is the angular acceleration in each case? Solution : a. The period of second-hand of the clock is T = 60 s, hence 2π 2π ω ω T 60 1 ω 0.11 rad s 36
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DF025 CHAPTER 8 Solution : b. The period of minute-hand of the clock is T = 60 min = 3600 s, hence 2π ω 3600 ω 1.74 10 3 rad s 1 c. The period of hour-hand of the clock is T = 12 h = 4.32 104 s, hence 2π ω 4.32 10 4 ω 1.45 10 4 rad s 1 d. The angular acceleration in each cases is zero. 37
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DF025 CHAPTER 8 Example 8 : 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 s1 and rolls in a straight line without slipping. If the rotation slows down with an angular acceleration of magnitude 1.90 rad s2, calculate the distance travelled by the coin before coming to rest. Solution : ω 18 rad s 1 0 ω 0 rad s 1 d 2.40 10 m2 α 1.90 rad s 2 s The radius of the coin is d r 1.20 10 2 m 2 38
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DF025 CHAPTER 8 Solution : The initial speed of the point at the edge the coin is u rω0 u 1.20 10 2 18 u 0.216 m s 1 1 and the final speed is v 0 m s The linear acceleration of the point at the edge the coin is given by a rα a 1.20 10 2 1.90 a 2.28 10 2 m s 2 Therefore the distance travelled by the coin is v u 2as 2 2 0 0.216 2 2.28 10 2 s 2 s 1.02 m 39
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DF025 CHAPTER 8 Exercise 8.1b&c : 1. A disk 8.00 cm in radius rotates at a constant rate of 1200 rev min-1 about its central axis. Determine a. its angular speed, b. the tangential speed at a point 3.00 cm from its centre, c. the radial acceleration of a point on the rim, d. the total distance a point on the rim moves in 2.00 s. ANS. : 126 rad s1; 3.77 m s1; 1.26 103 m s2; 20.1 m 2. A 0.35 m diameter grinding wheel rotates at 2500 rpm. Calculate a. its angular velocity in rad s1, b. the linear speed and the radial acceleration of a point on the edge of the grinding wheel. ANS. : 262 rad s1; 46 m s1, 1.2 104 m s2 40
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DF025 CHAPTER 8 Exercise 8.1b&c : 3. A rotating wheel required 3.00 s to rotate through 37.0 revolution. Its angular speed at the end of the 3.00 s interval is 98.0 rad s-1. Calculate the constant angular acceleration of the wheel. ANS. : 13.6 rad s2 4. A wheel rotates with a constant angular acceleration of 3.50 rad s2. a. If the angular speed of the wheel is 2.00 rad s1 at t =0, through what angular displacement does the wheel rotate in 2.00 s. b. Through how many revolutions has the wheel turned during this time interval? c. What is the angular speed of the wheel at t = 2.00 s? ANS. : 11.0 rad; 1.75 rev; 9.00 rad s1 41
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DF025 CHAPTER 8 Exercise 8.1b&c : 5. A bicycle wheel is being tested at a repair shop. The angular velocity of the wheel is 4.00 rad s-1 at time t = 0 , and its angular acceleration is constant and equal 1.20 rad s-2. A spoke OP on the wheel coincides with the +x-axis at time t = 0 as shown in Figure 7.8. y P x O Figure 8.8 a. What is the wheel’s angular velocity at t = 3.00 s? b. What angle in degree does the spoke OP make with the positive x-axis at this time? ANS. : 0.40 rad s1; 18 42
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DF025 CHAPTER 8 Learning Outcome: 8.2 Equilibrium of a uniform rigid body (2 hours) At the end of this chapter, students should be able to: a) Define and use torque, τ b) State and use conditions for equilibrium of rigid body: F x 0, F y 0, τ 0 Examples of problems : Fireman ladder leaning on a wall, see-saw, pivoted / suspended horizontal bar. Sign convention for moment or torque : +ve : anticlockwise 43 ve : clockwise
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DF025 CHAPTER 8 8.2 Equilibrium of a rigid body Non-concurrent forces is defined as the forces whose lines of action do not pass through a single common point. The forces cause the rotational motion on the body. The combination of concurrent and non-concurrent forces cause rolling motion on the body. (translational and rotational motion) Figure 5.11 shows an example of non-concurrent forces. F1 F2 F4 Figure 8.2 F3 44
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DF025 CHAPTER 8 Torque (moment of a force), The magnitude of the torque is defined as the product of a force and its perpendicular distance from the line of action of the force to the point (rotation axis). τ Fd OR where τ : magnitude of the torque F : magnitude of the force d : perpendicular distance (moment arm) Because of d r sin where r : distance between the pivot point (rotation axis) and the point of application of force. Thus Fr sin OR r F where : angle between F and r 45
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DF025 CHAPTER 8 It is a vector quantity. The dimension of torque is F d ML T2 2 The unit of torque is N m (newton metre), a vector product unlike the joule (unit of work), also equal to a newton metre, which is scalar product. Torque is occurred because of turning (twisting) effects of the forces on a body. Sign convention of torque: Positive - turning tendency of the force is anticlockwise. Negative - turning tendency of the force is clockwise. The value of torque depends on the rotation axis and the magnitude of applied force. 46
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DF025 CHAPTER 8 Case 1 : Consider a force is applied to a metre rule which is pivoted at one end as shown in Figures 5.12a and 5.12b. F τ Fd (anticlockwise) d Figure 8.12a Line of action of a force Pivot point (rotation axis) Point of action of a force d r sin θ F θ τ Fd Fr sin θ r (anticlockwise) Figure 8.12b 47
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DF025 CHAPTER 8 Case 2 : Consider three forces are applied to the metre rule which is pivoted at one end (point O) as shown in Figures 5.13. τ1 F1d1 F1r1 sin θ1 F3 d1 r1 sin θ1 τ 2 F2 d 2 F2 r2 sin θ2 F1 τ 3 F3 d 3 F3 r3 sin θ3 0 r2 r1 θ1 Therefore the resultant (nett) torque is Od 2 r2 sin θ2 θ2 τ O τ1 τ 2 τ 3 τ F2 Figure 8.13 O F1d1 F2 d 2 Caution : If the line of action of a force is through the rotation axis then τ Fr sin θ and θ 0 τ 0 Simulation 8.2 48
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DF025 CHAPTER 8 Example 4 : Determine a resultant torque of all the forces about rotation axis, O in the following problems. a. F 10 N 2 5m 5m F1 30 N 3m O 6m 3m 10 m F3 20 N 49
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DF025 CHAPTER 8 Example 4 : b. 10 m F1 30 N 3m 6m β O 3m F3 20 N 5m 5m α F4 25 N F2 10 N 50
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DF025 CHAPTER 8 Solution : F2 10 N a. 5m 5m F1 30 N d1 3 m 6m O d2 5 m 10 m Force Torque (N m), o=Fd=Frsin F3 20 N F1 30 3 90 F2 10 5 50 The resultant torque: F3 0τ O 90 50 40 N m (clockwise) 51
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DF025 CHAPTER 8 Solution : b. 10 m F1 30 N 3m d3 d 3 m β O 1 6m β r 5m 3 sin β 0.515 F3 20 N 5m 5m 32 5 2 F2 10 N α F4 25 NForce Torque (N m), o=Fd=Frsin F1 30 3 90 F2 0 The resultant torque: F3 F3 r sin β 2050.515 51.5 τ O 90 51.5 F4 0 τ O 38.5 N m (clockwise) 52
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DF025 CHAPTER 8 8.2 Equilibrium of a rigid body Rigid body is defined as a body with definite shape that doesn’t change, so that the particles that compose it stay in fixed position relative to one another even though a force is exerted on it. If the rigid body is in equilibrium, means the body is translational and rotational equilibrium. There are two conditions for the equilibrium of forces acting on a rigid body. The vector sum of all forces acting on a rigid body must be zero. F F nett 0 OR F x 0, Fy 0, F z 0 53
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DF025 CHAPTER 8 The vector sum of all external torques acting on a rigid body must be zero about any rotation axis. τ nett 0 This ensures rotational equilibrium. This is equivalent to the three independent scalar equations along the direction of the coordinate axes, τ x 0, τ y 0, τ z 0 Centre of gravity, CG is defined as the point at which the whole weight of a body may be considered to act. A force that exerts on the centre of gravity of an object will cause a translational motion. 54
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DF025 CHAPTER 8 Figures 5.14 and 5.15 show the centre of gravity for uniform (symmetric) object i.e. rod and sphere rod – refer to the midway point between its end. l CG l l 2 2 Figure 5.14 sphere – refer to geometric centre. CG Figure 5.15 55
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DF025 CHAPTER 8 5.3.4 Problem solving strategies for equilibrium of a rigid body The following procedure is recommended when dealing with problems involving the equilibrium of a rigid body: Sketch a simple diagram of the system to help conceptualize the problem. Sketch a separate free body diagram for each body. Choose a convenient coordinate axes for each body and construct a table to resolve the forces into their components and to determine the torque by each force. Apply the condition for equilibrium of a rigid body : F x 0; F y 0 and τ 0 Solve the equations for the unknowns. 56
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DF025 CHAPTER 8 Example 5 : A 35 O 75 B cm cm W1 W2 Figure 5.16 A hanging flower basket having weight, W2 =23 N is hung out over the edge of a balcony railing on a uniform horizontal beam AB of length 110 cm that rests on the balcony railing. The basket is counterbalanced by a body of weight, W1 as shown in Figure 5.16. If the mass of the beam is 3.0 kg, calculate a. the weight, W1 needed, b. the force exerted on the beam at point O. (Given g =9.81 m s2) 57
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DF025 CHAPTER 8 Solution : m 3 kg; W2 23 N N The free body diagram of the beam : 0.20 m A 0.35 m 0.75 m B CG O W2 0.55 m 0.55 m W1 mg Let point O as the rotation axis. Force y-comp. (N) Torque (N m), o=Fd=Frsin W1 W1 W1 0.75 0.75W1 W2 23 230.35 8.05 mg 39.81 29.4 0.20 5.88 29.4 N N 0 58
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DF025 CHAPTER 8 Solution : Since the beam remains at rest thus the system in equilibrium. a. Hence τ O 0 0.75W1 8.05 5.88 0 W1 2.89 N b. and F y 0 W1 23 29.4 N 0 2.89 23 29.4 N 0 N 55.3 N 59
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DF025 CHAPTER 8 Example 6 : A uniform ladder AB of length 10 m and mass 5.0 kg leans against a smooth wall as shown in Figure 5.17. The height of the A end A of the ladder is 8.0 m from the rough floor. a. Determine the horizontal and vertical forces the floor exerts on the end B of the ladder when a firefighter of mass 60 kg is 3.0 m from B. b. If the ladder is just on the verge of slipping when the firefighter is 7.0 m smooth wall B up the ladder , Calculate the coefficient of static friction between ladder and rough floor floor. Figure 5.17 (Given g =9.81 m s2) 60
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DF025 CHAPTER 8 Solution : ml 5.0 kg; m f 60 kg a. The free body diagram of the ladder : Let point B as the rotation axis. A N1 x-comp. y-comp. Torque (N m), 8Force (N) (N) B=Fd=Frsin α sin α 0.8 β 10 ml g 0 49.1 49.15.0sin β 6 sin β 0.6 147 10 mf g 0 589 589 3.0 sin β 8.0 m CG 10 m 1060 N1 N1 0 N1 10sin α ml g β 3.0 m 8 N 1 N2 mf g βN2 0 N2 0 5.0 m α B fs fs 0 0 fs 6.0 m 61
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DF025 CHAPTER 8 Solution : Since the ladder in equilibrium thus τ B 0 147 1060 8N1 0 N1 151 N F x 0 N1 f s 0 Horizontal force:f s 151 N F y 0 49.1 589 N 2 0 Vertical force: N 2 638 N 62
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DF025 CHAPTER 8 Solution : sin α 0.8; sin β 0.6 b. The free body diagram of the ladder : Let point B as the rotation axis. A N1 x-comp. y-comp. Torque (N m), αForce (N) (N) B=Fd=Frsin β ml g 0 49.1 49.15.0sin β 147 mf g β 10 m mf g 0 589 589 7.0sin β 8.0 m 7.0 m 2474 ml g β N1 N1 0 N1 10sin α 8 N 1 N2 5.0 m αN2 0 N2 0 B fs fs μs N 2 0 0 6.0 m 63
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DF025 CHAPTER 8 Solution : Consider the ladder stills in equilibrium thus τ B 0 147 2474 8N1 0 N1 328 N Fy 0 49.1 589 N 2 0 N 2 638 N F x 0 N 1 μs N 2 0 328 μs 638 0 μs 0.514 64
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DF025 CHAPTER 8 Example 7 : A floodlight of mass 20.0 kg in a park is supported at the end of a 10.0 kg uniform horizontal beam that is hinged to a pole as shown in Figure 5.18. A cable at an angle 30 with the beam helps to support the light. a. Sketch a free body diagram of the beam. b. Determine i. the tension in the cable, ii. the force exerted on the beam by the pole. Figure 5.18 (Given g =9.81 m s2) 65
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DF025 CHAPTER 8 Solution : m f 20.0 kg; mb 10.0 kg a. The free body diagram of the beam : S T 30 O 0.5l CG mb g l mf g b. Let point O as the rotation axis. Force x-comp. (N) y-comp. (N) Torque (N m), o=Fd=Frsin mf g 0 196 196 l mb g 0 98.1 98.10.5l 49.1l T T cos 30 T sin 30 Tl sin 30 0.5Tl S Sx Sy 0 66
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DF025 CHAPTER 8 Solution : b. The floodlight and beam remain at rest thus i. τ 0 O 196l 49.1l 0.5Tl 0 T 490 N ii. F x 0 T cos 30 S x 0 S x 424 N F y 0 196 98.1 T sin 30 S y 0 S y 49.1 N 67
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DF025 CHAPTER 8 Solution : b. ii. Therefore the magnitude of the force is S Sx S y 2 2 S 424 2 49.12 S 427 N and its direction is given by 1 Sy θ tan S x 1 49.1 θ tan 424 θ 6.61 from the +x-axis anticlockwise 68
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DF025 CHAPTER 8 Exercise 8.2 : Use gravitational acceleration, g = 9.81 m s2 1. F1 a B A F2 b D C γ Figure 8.19 F3 Figure 5.19 shows the forces, F1 =10 N, F2= 50 N and F3= 60 N are applied to a rectangle with side lengths, a = 4.0 cm and b = 5.0 cm. The angle is 30. Calculate the resultant torque about point D. ANS. : -3.7 N m 69
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DF025 CHAPTER 8 Exercise 8.2 : 2. Figure 5.20 A see-saw consists of a uniform board of mass 10 kg and length 3.50 m supports a father and daughter with masses 60 kg and 45 kg, respectively as shown in Figure 5.20. The fulcrum is under the centre of gravity of the board. Determine a. the magnitude of the force exerted by the fulcrum on the board, b. where the father should sit from the fulcrum to balance the system. ANS. : 1128 N; 1.31 m 70
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DF025 CHAPTER 8 Exercise 8.2 : 3. Figure 5.21 A traffic light hangs from a structure as show in Figure 5.21. The uniform aluminum pole AB is 7.5 m long has a mass of 8.0 kg. The mass of the traffic light is 12.0 kg. Determine a. the tension in the horizontal massless cable CD, b. the vertical and horizontal components of the force exerted by the pivot A on the aluminum pole. 71 ANS. : 248 N; 197 N, 248 N
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DF025 CHAPTER 8 Exercise 8.2 : 4. 30.0 cm 50.0 15.0 cm F Figure 5.22 A uniform 10.0 N picture frame is supported by two light string as shown in Figure 5.22. The horizontal force, F is applied for holding the frame in the position shown. a. Sketch the free body diagram of the picture frame. b. Calculate i. the tension in the ropes, ii. the magnitude of the horizontal force, F . ANS. : 1.42 N, 11.2 N; 7.20 N 72
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DF025 CHAPTER 8 8.3 Rotational Dynamics (1 hour) Centre of mass (CM) is defined as the point at which the whole mass of a body may be considered to be concentrated. Its coordinate (xCM, yCM) is given the expression below: n n mi xi mi yi i 1 xCM n ; yCM i 1 n mi i 1 mi i 1 th where mi : mass of the i particle xi : x coordinate of the i th particle yi : y coordinate of the i th particle 73
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DF025 CHAPTER 8 Example 9 : Two masses, 2 kg and 4 kg are located on the x-axis at x =2 m and x =5 m respectively. Determine the centre of mass of this system. Solution : m1 2 kg; m2 4 kg m 2 m1 CM x 0 2m 4m 5m 2 m x i i m1 x1 m2 x2 22 45 xCM i 1 xCM 2 m1 m2 2 4 m i 1 i xCM 4 m from x 0 OR 2 m from m1 74
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DF025 CHAPTER 8 Example 10 : A system consists of three particles have the following masses and coordinates : (1) 2 kg, (1,1) ; (2) 4 kg, (2,0) and (3) 6 kg, (2,2). Determine the coordinate of the centre of mass of the system. Solution : m1 1 kg; m2 2 kg; m3 3 kg The x coordinate of the CM is 3 m x i 1 i i m1 x1 m2 x2 m3 x3 xCM 3 m1 m2 m3 m i 1 i xCM 21 42 62 xCM 1.83 2 4 6 75
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DF025 CHAPTER 8 Solution : The y coordinate of the CM is 3 m y i 1 i i m1 y1 m2 y2 m3 y3 yCM 3 m1 m2 m3 y i 1 i yCM 21 40 62 yCM 1.17 2 4 6 Therefore the coordinate of the CM is 1.83,1.17 76
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DF025 CHAPTER 8 Moment of inertia, I Figure 7.9 shows a rigid body about a fixed axis O with angular velocity . m1 mn r1 rn r2 m2 Or m3 3 Figure 7.9 is defined as the sum of the products of the mass of each particle and the square of its respective distance from the rotation axis. 77
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DF025 CHAPTER 8 n OR I m1r12 m2 r22 m3 r32 ...mn rn2 m r i 1 i i 2 where I : moment of inertia of a rigid body about rotation axis m : mass of particle r : distance from the particle to the rotation axis It is a scalar quantity. Moment of inertia, I in the rotational kinematics is analogous to the mass, m in linear kinematics. The dimension of the moment of inertia is M L2. The S.I. unit of moment of inertia is kg m2. The factors which affect the moment of inertia, I of a rigid body: a. the mass of the body, b. the shape of the body, c. the position of the rotation axis. 78
79.
DF025 CHAPTER 8 Moments of inertia of various bodies Table 7.3 shows the moments of inertia for a number of objects about axes through the centre of mass. Shape Diagram Equation Hoop or ring or thin cylindrical CM I CM MR 2 shell 1 Solid cylinder or disk CM I CM MR 2 2 79
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DF025 CHAPTER 8 Moments of inertia of various bodies Table 7.3 shows the moments of inertia for a number of objects about axes through the centre of mass. Shape Diagram Equation Uniform rod or long thin rod with 1 rotation axis CM I CM ML2 through the 12 centre of mass. 2 Solid Sphere CM I CM MR 2 5 80
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DF025 CHAPTER 8 Moments of inertia of various bodies Table 7.3 shows the moments of inertia for a number of objects about axes through the centre of mass. Shape Diagram Equation Hollow Sphere or 2 thin spherical CM I CM MR 2 shell 3 Table 7.3 81
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DF025 CHAPTER 8 Example 11 : Four spheres are arranged in a rectangular shape of sides 250 cm and 120 cm as shown in Figure 7.10. 2 kg 3 kg 60 cm A B O 60 cm 5 kg 250 cm 4 kg Figure 7.10 The spheres are connected by light rods . Determine the moment of inertia of the system about an axis a. through point O, b. along the line AB. 82
83.
DF025 CHAPTER 8 Solution : m1 2 kg; m2 3 kg; m3 4 kg; m4 5 kg a. rotation axis about point O, m1 m2 r1 r2 0.6 m 1.25 m O r4 r3 m4 m3 Since r1= r2= r3= r4= r thus r 0.6 1.25 2 2 1.39 m and the connecting rods are light therefore I O m1r12 m2 r22 m3r32 m4 r42 I O r m1 m2 m3 m4 1.39 2 3 4 5 2 2 I O 27.0 kg m 2 83
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DF025 CHAPTER 8 Solution : m1 2 kg; m2 3 kg; m3 4 kg; m4 5 kg b. rotation axis along the line AB, m1 m2 r1 r2 A B r4 r3 m4 m3 r1= r2= r3= r4= r=0.6 m therefore I AB m1r12 m2 r22 m3r32 m4 r42 I AB r m1 m2 m3 m4 2 I AB 0.6 2 3 4 5 2 I AB 5.04 kg m 2 84
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DF025 CHAPTER 8 Parallel-Axis Theorem (Steiner’s Theorem) States that the moment of inertia, I about any axis parallel to and a distance, d away from the axis through the centre of mass, ICM is given by I I CM Md 2 where I : moment of inertia about a new rotation axis I CM : moment of inertia about an axis through CM M : mass of the rigid body d : distance between the new axis and the original axis 85
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DF025 CHAPTER 8 Example 12 : Determine the moment of inertia of a solid cylinder of radius R and mass M about an axis tangent to its edge and parallel to its symmetry axis as shown in Figure 7.11. d CM Figure 7.11 Given the moment of inertia of the solid cylinder about axis through the centre of mass is 1 I CM MR 2 2 86
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DF025 CHAPTER 8 Solution : CM CM 1 I CM MR 2 d Initial 2 Final From the diagram, d = R By using the parallel axis theorem, I I CM Md 2 1 I MR 2 MR 2 2 3 I MR 2 2 87
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DF025 CHAPTER 8 Torque, Relationship between torque, and angular acceleration, Consider a force, F acts on a rigid body freely pivoted on an axis through point O as shown in Figure 7.12. a1 m1 mn r1 F an rn O r2 a2 m2 Figure 7.12 The body rotates in the anticlockwise direction and a nett torque is produced. 88
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DF025 CHAPTER 8 A particle of mass, m1 of distance r1 from the rotation axis O will experience a nett force F1 . The nett force on this particle is F1 m1a1 and a1 r1α F1 m1r1α The torque on the mass m1 is 1 r1F1 sin 90 1 m1r1 2 The total (nett) torque on the rigid body is given by m1r12 m2 r22 ... mn rn2 n n i 1 mi ri 2 and mi ri 2 I i 1 I 89
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DF025 CHAPTER 8 From the equation, the nett torque acting on the rigid body is proportional to the body’s angular acceleration. Note : Nett torqu e , I is analogous to the Nett force, F ma 90
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DF025 CHAPTER 8 Example 13 : Forces, F1 = 5.60 N and F2 = 10.3 N are applied tangentially to a disc with radius 30.0 cm and the mass 5.00 kg as shown in Figure 7.13. F2 O 30.0 cm F1 Calculate, Figure 7.13 a. the nett torque on the disc. b. the magnitude of angular acceleration influence by the disc. 1 ( Use the moment of inertia, I CM MR 2 ) 2 91
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DF025 CHAPTER 8 Solution : R 0.30 m; M 5.00 kg a. The nett torque on the disc is 1 2 RF RF 1 R F1 F2 2 0.30 5.60 10.3 1.41 N m b. By applying the relationship between torque and angular acceleration, 1 2 I 2 MR 1 2 1.41 5.000.30 2 6.27 rad s 2 92
93.
DF025 CHAPTER 8 Example 14 : A wheel of radius 0.20 m is mounted on a frictionless horizontal axis. The moment of inertia of the wheel about the axis is 0.050 kg m2. A light string wrapped around the wheel is attached to a 2.0 kg block that slides on a horizontal frictionless surface. A horizontal force of magnitude P = 3.0 N is applied to the block as shown in Figure 7.14. Assume the string does not slip on the wheel. Figure 7.14 a. Sketch a free body diagram of the wheel and the block. b. Calculate the magnitude of the angular acceleration of the wheel. 93
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DF025 CHAPTER 8 Solution : R 0.20 m; I 0.050 kg m 2 ; P 3.0 N; m 2.0 kg a. Free body diagram : for wheel, T S W a for block, N T P Wb 94
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DF025 CHAPTER 8 Solution : R 0.20 m; I 0.050 kg m 2 ; P 3.0 N; m 2.0 kg b. For wheel, τ Iα Iα RT Iα T (1) For block, R F ma P T ma (2) By substituting eq. (1) into eq. (2), thus Iα P ma and a Rα R Iα P mRα R 0.050α 3.0 2.00.20 α α 4.62 rad s 2 0.20 95
96.
DF025 CHAPTER 8 Example 15 : An object of mass 1.50 kg is suspended from a rough pulley of radius 20.0 cm by light string as shown in Figure 7.15. The pulley has a moment of inertia 0.020 kg m2 about the axis of the pulley. The object is released from rest and the pulley rotates without encountering frictional force. Assume that R the string does not slip on the pulley. After 0.3 s, determine a. the linear acceleration of the object, b. the angular acceleration of the pulley, 1.50 kg c. the tension in the string, Figure 7.15 d. the liner velocity of the object, e. the distance travelled by the object. (Given g = 9.81 m s-2) 96
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DF025 CHAPTER 8 Solution : a. Free body diagram : for pulley, S τ Iα a RT Iα and α a R RT I R T W Ia T 2 (1) for block, T R F ma a mg T ma (2) mg 97
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DF025 CHAPTER 8 Solution : R 0.20 m; I 0.020 kg m 2 ; m 1.50 kg; u 0; t 0.3 s a. By substituting eq. (1) into eq. (2), thus Ia mg 2 ma R 0.020 a 1.50 9.81 1.50 a 0.20 2 a 7.36 m s 2 b. By using the relationship between a and , hence a Rα 7.36 0.20α α 36.8 rad s 2 98
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DF025 CHAPTER 8 Solution : R 0.20 m; I 0.020 kg m 2 ; m 1.50 kg; u 0; t 0.3 s c. From eq. (1), thus Ia T 2 T 0.020 7.36 R 0.20 2 T 3.68 N d. By applying the equation of liner motion, thus v u at v 0 7.36 0.3 v 2.21 m s 1 (downwards) e. The distance travelled by the object in 0.3 s is 1 2 s ut at 2 s 0 7.36 0.3 1 s 0.331 m 2 2 99
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DF025 CHAPTER 8 Exercise 8.3 : Use gravitational acceleration, g = 9.81 m s2 1. Three odd-shaped blocks of chocolate have following masses and centre of mass coordinates: 0.300 kg, (0.200 m,0.300 m); 0.400 kg, (0.100 m. -0.400 m); 0.200 kg, (-0.300 m, 0.600 m). Determine the coordinates of the centre of mass of the system of three chocolate blocks. ANS. : (0.044 m, 0.056 m) 2. Figure 7.16 shows four masses that are held at 70 g the corners of a square by a very light 40 cm frame. Calculate the moment of inertia 80 cm of the system about an axis perpendicular B to the plane 150 g A 150 g a. through point A, and 80 cm b. through point B. Figure 7.16 70 g ANS. : 0.141 kg m2; 0.211 kg m2 100
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DF025 CHAPTER 8 Exercise 8.3 : 3. A 5.00 kg object placed on a frictionless horizontal table is connected to a string that passes 2.00 m s 2 over a pulley and then is fastened to a hanging 9.00 kg object as in T1 Figure 7.17. The pulley has a radius of 0.250 m and moment of inertia I. The block on the table is T2 moving with a constant acceleration of 2.00 m s2. a. Sketch free body diagrams of both objects and pulley. b. Calculate T1 and T2 the tensions Figure 7.17 in the string. c. Determine I. ANS. : 10.0 N, 70.3 N; 1.88 kg m2 101
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DF025 CHAPTER 8 Learning Outcome: 8.4 Work and Energy of Rotational Motion(2 hours) At the end of this chapter, students should be able to: a) Solve problems related to: Rotational kinetic energy, 1 2 K r Iω 2 work, W τθ power, P τω 102
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DF025 CHAPTER 8 Rotational kinetic energy and power Rotational kinetic energy, Kr Consider a rigid body rotating about the axis OZ as shown in Figure 7.18. Z v1 m1 mn r1 vn rn r2 v2 O m2 r3 v3 m3 Figure 7.18 Every particle in the body is in the circular motion about point O. 103
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DF025 CHAPTER 8 The rigid body has a rotational kinetic energy which is the total of kinetic energy of all the particles in the body is given by 1 1 1 1 K r m1v1 m2 v2 m3v3 ... mn vn 2 2 2 2 2 2 2 2 1 1 1 1 K r m1r1 ω m2 r2 ω m3 r3 ω ... mn rn2 ω2 2 2 2 2 2 2 2 2 2 2 1 2 K r ω m1r12 m2 r22 m3 r32 ... mn rn2 2 1 2 n n 2 K r ω mi ri and mi ri I 2 2 i 1 i 1 1 2 K r Iω 2 104
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DF025 CHAPTER 8 From the formula for translational kinetic energy, Ktr 1 2 K tr mv 2 After comparing both equations thus is analogous to v I is analogous to m For rolling body without slipping, the total kinetic energy of the body, K is given by K K tr K r where K tr : translati onal kinetic energy K r : rotational kinetic energy 105
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DF025 CHAPTER 8 Example 16 : A solid sphere of radius 15.0 cm and mass 10.0 kg rolls down an inclined plane make an angle 25 to the horizontal. If the sphere rolls without slipping from rest to the distance of 75.0 cm and the inclined surface is smooth, calculate a. the total kinetic energy of the sphere, b. the linear speed of the sphere, c. the angular speed about the centre of mass. 2 (Given the moment of inertia of solid sphere is I CM mR 2and the gravitational acceleration, g = 9.81 m s2) 5 106
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DF025 CHAPTER 8 Solution : R 0.15 m; m 10.0 kg s 0.75 m R h s sin 25 v CM 25 a. From the principle of conservation of energy, E E i f mgh K K mgs sin 25 K 10.0 9.810.75sin 25 K 31.1 J 107
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DF025 CHAPTER 8 Solution : R 0.15 m; m 10.0 kg b. The linear speed of the sphere is given by K K tr K r 1 2 1 2 v K mv Iω and ω 2 2 R 2 1 2 12 2 v K mv mR 2 25 R 7 K mv 2 10 31.1 10.0 v 2 7 v 2.11 m s 1 10 c. By using the relationship between v and , thus v Rω 2.11 0.15ω ω 14.1 rad s 1 108
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DF025 CHAPTER 8 Example 17 : The pulley in the Figure 7.19 has a radius of 0.120 m and a moment of inertia 0.055 g cm2. The rope does not slip on the pulley rim. Calculate the speed of the 5.00 kg block just before it strikes the floor. (Given g = 9.81 m s2) 5.00 kg 7.00 m 2.00 kg Figure 7.19 109
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DF025 CHAPTER 8 Solution : m1 5.00 kg; m2 2.00 kg; R 0.120 m; h 7.00 m The moment of inertia of the pulley, 10 3 kg 10 4 m 2 I 0.055 g 1 cm2 1 g 1 cm2
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