# EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS

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1 EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS Today s Objectives: Students will be able to: 1. Analyze the planar kinetics In-Class Activities: of a rigid body undergoing rotational motion. Check Homework Reading Quiz Applications Rotation About an Axis Equations of Motion Concept Quiz Group Problem Solving Attention Quiz READING QUIZ 1. In rotational motion, the normal component of acceleration at the body s center of gravity (G) is always A) zero. B) tangent to the path of motion of G. C) directed from G toward the center of rotation. D) directed from the center of rotation toward G. 2. If a rigid body rotates about point O, the sum of the moments of the external forces acting on the body about point O equals A) I G α B) I O α C) m a G D) m a O 1

2 APPLICATIONS The crank on the oil-pump rig undergoes rotation about a fixed axis, caused by the driving torque M from a motor. As the crank turns, a dynamic reaction is produced at the pin. This reaction is a function of angular velocity, angular acceleration, and the orientation of the crank. Pin at the center of rotation. If the motor exerts a constant torque M on the crank, does the crank turn at a constant angular velocity? Is this desirable for such a machine? APPLICATIONS The Catherine wheel is a fireworks display consisting of a coiled tube of powder pinned at its center. As the powder burns, the mass of powder decreases as the exhaust gases produce a force directed tangent to the wheel. This force tends to rotate the wheel. If the powder burns at a constant rate, the exhaust gases produce a constant thrust. Does this mean the angular acceleration is also constant? Why or why not? What is the resulting effect on the fireworks display? 2

3 EQUATIONS OF MOTION FOR PURE ROTATION (Section 17.4) When a rigid body rotates about a fixed axis perpendicular to the plane of the body at point O, the body s center of gravity G moves in a circular path of radius r G. Thus, the acceleration of point G can be represented by a tangential component (a G ) t = r G α and a normal component (a G ) n = r G ω 2. Since the body experiences an angular acceleration, its inertia creates a moment of magnitude I G α equal to the moment of the external forces about point G. Thus, the scalar equations of motion can be stated as: F n = m (a G ) n = m r G ω 2 F t = m (a G ) t = m r G α M G = I G α EQUATIONS OF MOTION Note that the M G moment equation may be replaced by a moment summation about any arbitrary point. Summing the moment about the center of rotation O yields M O = I G α + r G m (a G ) t = (I G + m (r G ) 2 ) α From the parallel axis theorem, I O = I G + m(r G ) 2, therefore the term in parentheses represents I O. Consequently, we can write the three equations of motion for the body as: F n = m (a G ) n = m r G ω 2 F t = m (a G ) t = m r G α M O = I O α 3

4 PROCEDURE FOR ANALYSIS Problems involving the kinetics of a rigid body rotating about a fixed axis can be solved using the following process. 1. Establish an inertial coordinate system and specify the sign and direction of (a G ) n and (a G ) t. 2. Draw a free body diagram accounting for all external forces and couples. Show the resulting inertia forces and couple (typically on a separate kinetic diagram). 3. Compute the mass moment of inertia I G or I O. 4. Write the three equations of motion and identify the unknowns. Solve for the unknowns. 5. Use kinematics if there are more than three unknowns (since the equations of motion allow for only three unknowns). EXAMPLE Given:A rod with mass of 20 kg is rotating at 5 rad/s at the instant shown. A moment of 60 N m is applied to the rod. Find: The angular acceleration α and the reaction at pin O when the rod is in the horizontal position. Plan: Since the mass center, G, moves in a circle of radius 1.5 m, it s acceleration has a normal component toward O and a tangential component acting downward and perpendicular to r G. Apply the problem solving procedure. 4

5 Solution: EXAMPLE FBD & Kinetic Diagram Equations of motion: + F n = ma n = mr G ω 2 Using I G = (ml 2 )/12 and r G = (0.5)(l), we can write: M O = α[(ml 2 /12) + (ml 2 /4)] = (ml 2 /3)α where (ml 2 /3) = I O. After substituting: (9.81)(1.5) = 20(3 2 /3)α O n = 20(1.5)(5) 2 = 750 N + F t = ma t = mr G α -O t + 20(9.81) = 20(1.5)α + M O = I G α + m r G α (r G ) Solving: α = 5.9 rad/s 2 O t = 19 N CONCEPT QUIZ 1. If a rigid bar of length l (above) is released from rest in the horizontal position (θ = 0), the magnitude of its angular acceleration is at maximum when A) θ = 0 B) θ = 90 C) θ = 180 D) θ = 0 and In the above problem, when θ = 90, the horizontal component of the reaction at pin O is A) zero B) m g C) m (l/2) ω 2 D) None of the above. O l θ 5

6 GROUP PROBLEM SOLVING Given: W disk = 15 lb, W rod = 10 lb, ω = 8 rad/s at this instant. Find: The horizontal and vertical components of the reaction at pin O when the rod is horizontal. Plan: Draw the free body diagram and kinetic diagram of the rod and disk as one unit. Then apply the equations of motion. Solution: 15 lb 10 lb GROUP PROBLEM SOLVING m d (3.75)(8) 2 (I G ) r α (I G ) d α mr (1.5)(8)2 = O x O y Equations of motion: m d (3.75α) m r (1.5α) F x = m(a G ) x : O x = (15/32.2)(3.75)(8) 2 + (10/32.2)(1.5)(8) 2 O x = 142 lb F y = m(a G ) y : O y = -(15/32.2)(3.75α) (10/32.2)(1.5α) M O = I o α: 15(3.75) + 10(1.5) = [0.5(15/32.2)(0.75) 2 + (15/32.2)(3.75) 2 ] disk α Therefore, + [(1/12)(10/32.2)(3) 2 α = 9.36 rad/s 2, O + (10/32.2)(1.5) 2 y = 4.29 lb ] rod α 6

7 (a) m ATTENTION QUIZ (b) m 10 lb 10 lb T 1. A drum of mass m is set into motion in two ways: (a) by a constant 10 lb force, and, (b) by a block of weight 10 lb. If α a and α b represent the angular acceleration of the drum in each case, select the true statement. A) α a > α b B) α a < α b C) α a = α b D) None of the above. 2. For the same problem, the tension T in the cable in case (b) is A) T = 10 lb B) T < 10 lb C) T > 10 lb D) None of the above. EQUATIONS OF MOTION: GENERAL PLANE MOTION Today s Objectives: Students will be able to: 1. Analyze the planar kinetics of a rigid body undergoing general plane motion. In-Class Activities: Check Homework Reading Quiz Applications Equations of Motion Frictional Rolling Problems Concept Quiz Group Problem Solving Attention Quiz 7

8 READING QUIZ 1. If a disk rolls on a rough surface without slipping, the acceleration af the center of gravity (G) will and the friction force will be. A) not be equal to α r ; B) be equal to α r ; less than μ s N equal to μ k N C) be equal to α r ; less than μ s N D) None of the above. 2. If a rigid body experiences general plane motion, the sum of the moments of external forces acting on the body about any point P is equal to A) I P α. B) I P α + ma P. C) ma G. D) I G α + r GP x ma P. APPLICATIONS As the soil compactor accelerates forward, the front roller experiences general plane motion (both translation and rotation). What are the loads experienced by the roller shaft or bearings? = The forces shown on the roller s FBD cause the accelerations shown on the kinetic diagram. 8

9 APPLICATIONS During an impact, the center of gravity of this crash dummy will decelerate with the vehicle, but also experience another acceleration due to its rotation about point A. How can engineers use this information to determine the forces exerted by the seat belt on a passenger during a crash? GENERAL PLANE MOTION (Section 17.5) When a rigid body is subjected to external forces and couple-moments, it can undergo both translational motion as well as rotational motion. This combination is called general plane motion. Using an x-y inertial coordinate system, the equations of motions about the center of mass, G, may be written as F x = m (a G ) x P F y = m (a G ) y M G = I G α 9

10 GENERAL PLANE MOTION Sometimes, it may be convenient to write the moment equation about some point P other than G. Then the equations of motion are written as follows. F x = m (a G ) x F y = m (a G ) y M P = (M k ) P P In this case, (M k ) P represents the sum of the moments of I G α and ma G about point P. FRICTIONAL ROLLING PROBLEMS When analyzing the rolling motion of wheels, cylinders, or disks, it may not be known if the body rolls without slipping or if it slides as it rolls. For example, consider a disk with mass m and radius r, subjected to a known force P. The equations of motion will be F x = m(a G ) x => P - F = ma G F y = m(a G ) y => N - mg = 0 M G = I G α => F r = I G α There are 4 unknowns (F, N, α, and a G )in these three equations. 10

11 FRICTIONAL ROLLING PROBLEMS Hence, we have to make an assumption to provide another equation. Then we can solve for the unknowns. The 4 th equation can be obtained from the slip or non-slip condition of the disk. Case 1: Assume no slipping and use a G = α r as the 4 th equation and DO NOT use F f = μ s N. After solving, you will need to verify that the assumption was correct by checking if F f μ s N. Case 2: Assume slipping and use F f = μ k Nas the 4 th equation. In this case, a G αr. PROCEDURE FOR ANALYSIS Problems involving the kinetics of a rigid body undergoing general plane motion can be solved using the following procedure. 1. Establish the x-y inertial coordinate system. Draw both the free body diagram and kinetic diagram for the body. 2. Specify the direction and sense of the acceleration of the mass center, a G, and the angular acceleration α of the body. If necessary, compute the body s mass moment of inertia I G. 3. If the moment equation ΣM p = Σ(M k ) p is used, use the kinetic diagram to help visualize the moments developed by the components m(a G ) x, m(a G ) y, and I G α. 4. Apply the three equations of motion. 11

12 PROCEDURE FOR ANALYSIS 5. Identify the unknowns. If necessary (i.e., there are four unknowns), make your slip-no slip assumption (typically no slipping, or the use of a G =αr, is assumed first). 6. Use kinematic equations as necessary to complete the solution. 7. If a slip-no slip assumption was made, check its validity!!! Key points to consider: 1. Be consistent in assumed directions. The direction of a G must be consistent with α. 2. If F f = μ k N is used, F f must oppose the motion. As a test, assume no friction and observe the resulting motion. This may help visualize the correct direction of F f. EXAMPLE Given: A spool has a mass of 8 kg and a radius of gyration (k G ) of 0.35 m. Cords of negligible mass are wrapped around its inner hub and outer rim. There is no slipping. Find: The angular acceleration (α) of the spool. Plan: Focus on the spool. Follow the solution procedure (draw a FBD, etc.) and identify the unknowns. 12

13 EXAMPLE Solution: The moment of inertia of the spool is I G = m (k G ) 2 = 8 (0.35) 2 = kg m 2 Method I Equations of motion: F y = m (a G ) y T = 8 a G M G = I G α 100 (0.2) T(0.5) = 0.98 α There are three unknowns, T, a G, α. We need one more equation to solve for 3 unknowns. Since the spool rolls on the cord at point A without slipping, a G = αr. So the third equation is: a G = 0.5α Solving these three equations, we find: α =10.3 rad/s 2, a G = 5.16 m/s 2, T = 19.8 N EXAMPLE Method II Now, instead of using a moment equation about G, a moment equation about A will be used. This approach will eliminate the unknown cord tension (T). Σ M A = Σ (M k ) A : 100 (0.7) (0.5) = 0.98 α + (8 a G )(0.5) Using the non-slipping condition again yields a G = 0.5α. Solving these two equations, we get α = 10.3 rad/s 2, a G = 5.16 m/s 2 13

14 CONCEPT QUIZ 1. An 80 kg spool (k G = 0.3 m) is on a rough surface and a cable exerts a 30 N load to the right. The friction force at A acts to the and the a G should be directed to the. A) right, left B) left, right C) right, right D) left, left 0.2m G 0.75m A 2. For the situation above, the moment equation about G is: A) 0.75 (F fa ) - 0.2(30) = - (80)(0.3 2 )α B) -0.2(30) = - (80)(0.3 2 )α C) 0.75 (F fa ) - 0.2(30) = - (80)(0.3 2 )α + 80a G D) None of the above. α 30N GROUP PROBLEM SOLVING Given:A 50 lb wheel has a radius of gyration k G = 0.7 ft. Find: The acceleration of the mass center if M = 35 lb-ft is applied. μ s = 0.3, μ k = Plan: Follow the problem solving procedure. Solution: The moment of inertia of the wheel about G is I G = m(k G ) 2 = (50/32.2)(0.7) 2 = slug ft 2 14

15 FBD: GROUP PROBLEM SOLVING Equations of motion: F x = m(a G ) x F A = (50/32.2) a G F y = m(a G ) y N A 50 = 0 M G = I G α F A = α We have 4 unknowns: N A, F A, a G and α. Another equation is needed before solving for the unknowns. GROUP PROBLEM SOLVING The three equations we have now are: F A = (50/32.2) a G N A - 50 = 0 35 (1.25)F A = α First, assume the wheel is not slipping. Thus, we can write a G = r α = 1.25 α Now solving the four equations yields: N A = 50.0 lb, F A = 21.3 lb, α = 11.0 rad/s 2, a G = 13.7 ft/s 2 The no-slip assumption must be checked. Is F A = 21.3 lb μ s N A = 15 lb? No! Therefore, the wheel slips as it rolls. 15

16 GROUP PROBLEM SOLVING Therefore, a G = 1.25α is not a correct assumption. The slip condition must be used. Again, the three equations of motion are: F A = (50/32.2) a G N A 50 = 0 35 (1.25)F A = α Since the wheel is slipping, the fourth equation is F A = μ k N A = 0.25 N A Solving for the unknowns yields: N A = 50.0 lb F A = 0.25 N A = 12.5 lb α = 25.5 rad/s 2 a G = 8.05 ft/s 2 ATTENTION QUIZ 1. A slender 100 kg beam is suspended by a cable. The moment equation about point A is A) 3(10) = 1/12(100)(4 2 ) α B) 3(10) = 1/3(100)(4 2 ) α 3m C) 3(10) = 1/12(100)(4 2 ) α + (100 a Gx )(2) 10 N D) None of the above. A 4m 2. Select the equation that best represents the no-slip assumption. A) F f = μ s N B) F f = μ k N C) a G = r α D) None of the above. 16

17 17

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