EQUATIONS OF MOTION: GENERAL PLANE MOTION Today’s Objectives: Students will be able to: 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

A) not be equal to a r; less than sN B) be equal to a r; equal to kN 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 a r; less than sN B) be equal to a r; equal to kN C) be equal to a r; less than sN 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) IP a B) IP a + maP C) m aG D) IG a + rGP × maP Answers: C A

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. Is the point A the IC? =

APPLICATIONS (continued) The lawn roller is pushed forward with a force of 200 N when the handle is at 45°. How can we determine its translation acceleration and angular acceleration? Does the acceleration depend on the coefficient’s of static and kinetic friction?

APPLICATIONS (continued) 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. Why? How can engineers use this information to determine the forces exerted by the seat belt on a passenger during a crash?

EQUATIONS OF MOTION: GENERAL PLANE MOTION (Section 17.5) When a rigid body is subjected to external forces and couple-moments, it can undergo both translational motion and rotational motion. This combination is called general plane motion.  Fx = m (aG)x  Fy = m (aG)y  MG = IG a Using an x-y inertial coordinate system, the equations of motions about the center of mass, G, may be written as:

EQUATIONS OF MOTION: GENERAL PLANE MOTION (continued) Sometimes, it may be convenient to write the moment equation about a point P other than G. Then the equations of motion are written as follows:  Fx = m (aG)x  Fy = m (aG)y  MP =  (Mk )P In this case,  (Mk )P represents the sum of the moments of IGa and maG 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:  Fx = m(aG)x => P  F = m aG  Fy = m(aG)y => N  mg = 0  MG = IGa => F r = IG a There are 4 unknowns (F, N, a, and aG) in these three equations.

FRICTIONAL ROLLING PROBLEMS (continued) Hence, we have to make an assumption to provide another equation. Then, we can solve for the unknowns. The 4th equation can be obtained from the slip or non-slip condition of the disk. Case 1: Assume no slipping and use aG = a r as the 4th equation and DO NOT use Ff = sN. After solving, you will need to verify that the assumption was correct by checking if Ff  sN. Case 2: Assume slipping and use Ff = kN as the 4th equation. In this case, aG  ar.

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, aG, and the angular acceleration a of the body. If necessary, compute the body’s mass moment of inertia IG. 3. If the moment equation Mp= (Mk)p is used, use the kinetic diagram to help visualize the moments developed by the components m(aG)x, m(aG)y, and IGa. 4. Apply the three equations of motion.

PROCEDURE FOR ANALYSIS (continued) 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 aG = a r, 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 using the assumed directions. The direction of aG must be consistent with a. 2. If Ff = kN is used, Ff must oppose the motion. As a test, assume no friction and observe the resulting motion. This may help visualize the correct direction of Ff.

EXAMPLE Given: A spool has a mass of 200 kg and a radius of gyration (kG) of 0.3 m. The coefficient of kinetic friction between the spool and the ground is k = 0.1. Find: The angular acceleration (a) of the spool and the tension in the cable. Plan: Focus on the spool. Follow the solution procedure (draw a FBD, etc.) and identify the unknowns.

= EXAMPLE (continued) Solution: The free body diagram and kinetic diagram for the body are: IG a maG = 1962 N Equations of motion: Fy = m (aG)y : NB − 1962 = 0  NB = 1962 N

EXAMPLE (continued) Note that aG = (0.4) a. Why ? Fx = m (aG)x: T – 0.1 NB = 200 aG = 200 (0.4) a  T – 196.2 = 80 a MG = IG a : 450 – T(0.4) – 0.1 NB (0.6) = 20 (0.3)2 a  450 – T(0.4) – 196.2 (0.6) = 1.8 a Solving these two equations, we get a = 7.50 rad/s2, T = 797 N

A) right, left B) left, right C) right, right D) left, left  CONCEPT QUIZ An 80 kg spool (kG = 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 aG should be directed to the __________ . A) right, left B) left, right C) right, right D) left, left 30N A 0.2m 0.75m  G • 2. For the situation above, the moment equation about G is? A) 0.75 (FfA) - 0.2(30) = - (80)(0.32) B) -0.2(30) = - (80)(0.32) C) 0.75 (FfA) - 0.2(30) = - (80)(0.32) + 80aG D) None of the above Answers: C A (Positive has been set to be CCW)

GROUP PROBLEM SOLVING Given: A 80 kg lawn roller has a radius of gyration of kG = 0.175 m. It is pushed forward with a force of 200 N. Find: The angular acceleration if s = 0.12 and k = 0.1. Plan: Follow the problem solving procedure. Solution: The moment of inertia of the roller about G is IG = m(kG)2 = (80)(0.175)2 = 2.45 kg·m2

GROUP PROBLEM SOLVING (continued) FBD: Equations of motion:  Fx = m(aG)x FA – 200 cos 45 = 80 aG  Fy = m(aG)y NA – 784.8 – 200 sin 45 = 0  MG = IG a – 0.2 FA = 2.45 a We have 4 unknowns: NA, FA, aG and a. Another equation is needed to allow solving for the unknowns.

GROUP PROBLEM SOLVING (continued) The three equations we have now are: FA – 200 cos 45° = 80 aG NA – 784.8 – 200 sin 45° = 0 – 0.2 FA= 2.45 a First, assume the wheel is not slipping. Thus, we can write aG = r a = 0.2 a Now solving the four equations yields: NA = 926.2 N, FA = 61.4 N, a = -5.01 rad/s2 , aG = -1.0 m/s2 The no-slip assumption must be checked. Is FA = 61.4 N  s NA= 111.1 N ? Yes, therefore, the wheel rolls without slip.

2. Select the equation that best represents the “no-slip” assumption. 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)(42)  B) 3(10) = 1/3(100)(42)  C) 3(10) = 1/12(100)(42)  + (100 aGx)(2) D) None of the above 4m 10 N 3m A 2. Select the equation that best represents the “no-slip” assumption. A) Ff = s N B) Ff = k N C) aG = r a D) None of the above Answers: 1. C 2. C

End of the Lecture Let Learning Continue