Physics 121 April 1 2008 Equilibrium Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 April 1 2008 Course Information Discussion of Exam 2 Topics to be discussed today Requirements for Equilibrium Gravitational Equilibrium Sample problems Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 April 1 2008 Homework set 7 is due on Saturday morning April 5 at 8 30 am This assignment has two components WeBWorK 75 Video analysis 25 Homework set 8 will be available later this week This assignment will be due on Saturday morning April 12 at 8 30 am Exam 2 will be returned in workshops this week Frank L H Wolfs Department of Physics and Astronomy University of Rochester Midterm Exam 2 Results Frank L H Wolfs Department of Physics and Astronomy University of Rochester Midterm Exam 2 Results Frank L H Wolfs Department of Physics and Astronomy University of Rochester Midterm Exam 2 Results Frank L H Wolfs Department of Physics and Astronomy University of Rochester Midterm Exam 2 Results Frank L H Wolfs Department of Physics and Astronomy University of Rochester Midterm Exam 2 Results Frank L H Wolfs Department of Physics and Astronomy University of Rochester Midterm Exam 2 What now What do you learn from your results up to now Exam 1 and Exam 2 60 everything is OK Exam 1 and Exam 2 40 you need help Please set up a time to meet with me Things are not going to get easier 40 Exam 1 and Exam 2 60 you probably will pass the course but changing your work habits might result in a better grade Look at the exam and its solutions and determine what you are missing e g do you have a problem applying the correct approach do you have problems working with variables etc Note even those students with 0 on Exam 1 and Exam 2 can still pass the course with a B or A But you need to act now Frank L H Wolfs Department of Physics and Astronomy University of Rochester Equilibrium An object is in equilibrium is the following conditions are met Net force 0 N first condition for equilibrium and Net torque 0 Nm second condition for equilibrium Note both conditions must be satisfied Even if the net force is 0 N the system can start to rotate if the net torque is not equal to 0 Nm Frank L H Wolfs Department of Physics and Astronomy University of Rochester Static Equilibrium What happens when the net force is equal to 0 N P constant What happens when the net torque is equal to 0 Nm L constant We conclude that an object in equilibrium can still move with constant linear velocity and rotate with constant angular velocity Conditions for static equilibrium P 0 kg m s L 0 kg m2 s Frank L H Wolfs Department of Physics and Astronomy University of Rochester Equilibrium Summary of conditions Equilibrium in 3D F F F x 0 y 0 and z 0 x 0 y 0 z 0 Equilibrium in 2D F F x 0 y 0 z 0 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Equilibrium Be sure to include all forces When evaluating conditions for equilibrium you need to make sure to include all forces acting on the system In the system shown in the Figure there are more forces acting on the system than the forces indicated For example there should be an upward force to balance the downward forces Of course the problem is how to apply the equilibrium conditions correctly Frank L H Wolfs Department of Physics and Astronomy University of Rochester Equilibrium The force of gravity Consider an extended rigid object that can rotate around a specific rotation point If the rotation point coincides with the center of gravity of the object it will be in static equilibrium in any orientation F m r m g What is the relation between the position of the center of mass and the position of the center of gravity Frank L H Wolfs Department of Physics and Astronomy University of Rochester Equilibrium The force of gravity F If the object is in equilibrium the net torque and the net force acting on it must be equal to 0 The net force acting on the object is equal to m r m g F F mg j F g m j F Mg j If the net force is equal to 0 N we must require that F Mg Frank L H Wolfs Department of Physics and Astronomy University of Rochester Equilibrium The force of gravity F The condition that F Mg is not sufficient for static equilibrium We must also require that the net torque is equal to 0 Nm The net torque acting on the object is equal to r m g m r m g m r g M r g cm If the net torque must be 0 Nm we must require that Mrcm g 0 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Equilibrium The force of gravity The system will be in equilibrium if Mrcm g 0 F The requires that m The center of gravity is located exactly below or above the rotation axis rcm parallel to vertical axis or The center of gravity coincides with the rotation axis rcm 0 Frank L H Wolfs r m g Used to determine the location of the center of gravity of an object Department of Physics and Astronomy University of Rochester Equilibrium Sample problem 1 A uniform beam of length L whose mass is m rest with its ends on two digital scales A block whose mass is M rests on the beam its center one fourth away from the beam s left end What do the scales read If the system is in equilibrium the net force must be 0 N F y Frank L H Wolfs Fl Fr L Mg mg Fl Fr Mg mg 0 Department of Physics and Astronomy University of Rochester Equilibrium Sample problem 1 If the system is in equilibrium the net torque must be 0 Nm Note the toque associated with a Fl force depends on the choice of the origin The condition that the torque must be 0 Nm must be satisfied with respect to any choice of origin If we choose the left scale as our origin the left scale force does not appear in our torque equation L L z Fl 0 Fr L Mg 4 mg 2 0 Frank L H Wolfs Fr L Mg mg Department of Physics and Astronomy University of Rochester Equilibrium Sample problem 1 The force generated by the right scale is thus equal to Fr Mg L L mg 4 2 1 Mg 1 mg L 4 2 We can now use the first condition of equilibrium to determine the force generated by the left scale 3 1 Fl Mg mg Fr Mg mg 4 2 Frank L H Wolfs Fl Fr L Mg mg Department of Physics and Astronomy University of Rochester Equilibrium Sample problem 2 …