Reading: Systems of Particles, Rotations 1, 2. Key concepts: Center of mass, momentum, motion relative to CM, collisions; vector product, kinetic energy of rotation, moment of inertia; torque, rotational 2nd law, rolling. 1.! Some questions about frictionless ice. A pole is standing on end on frictionless ice as shown. Its CM is at its midpoint. A slight nudge makes it slip and fall. On the drawing sketch its configuration when it is lying on the ice, and explain your choice. Batman and Robin have been stranded by one of the arch-fiends in the middle of a lake frozen with frictionless ice. Give a method by which they can reach the shore. A child is standing at the front of a large raft resting on frictionless ice. She walks to the back, then stops. Does the final location of the raft depend on how fast she walks? Explain. 2.! Two balls, each of mass m, are propelled at the same time from the same starting point. Ball A moves on a frictionless horizontal surface, starting with speed v 0. Ball B is thrown into 2v 0 v 0 the air with initial speed 2v 0, at elevation angle 60.! e.! What is the initial velocity (magnitude and direction) of the CM? Ans: v CM (0) = ( 7 /2) v 0 ; tan 1 ( 3 /2) 41 above horizontal. What is the acceleration (magnitude and direction) of the CM? [What is the total external force?] Ans: 1 2 g. Sketch on the drawing the trajectories of the balls and of the CM, while ball B is in flight. Indicate on your sketch the location of the CM at the instant when ball B has its maximum height.
3.! You are sitting in a cart at rest that can roll freely, facing the back. You have two large bricks in the cart, each of mass m. The mass of you plus the cart is M. You throw one brick off the back at speed v 0, and later throw the other off at the same speed relative to the cart. What is the final speed of you and the cart? A second time, starting at rest, you throw both bricks off at the same time with speed v 0. Now what is the final speed of you and the cart? Explain the difference between your answers. 4.! A proton of mass m and speed v 0 collides elastically with a deuteron of mass 2m initially at rest. The final speed of the proton is twice that of the deuteron. Let the incoming proton move along the x-axis. Draw a picture showing the situation after the collision, with the proton s final velocity making angle θ with the x-axis and the deuteron s final velocity making angle φ with that axis. Find the final speeds of the two particles. Ans: v d = v 0 / 6, v p = 2v 0 / 6. Find the two angles. [The answer is numerical.] Ans: Both are cos 1 ( 6 /4). 5.! A pair of masses are connected by a massless rigid rod of length l as shown. This system is to be rotated about a fixed axis perpendicular to the rod. If the axis passes through the CM, what is the moment of inertia? Ans: 2 3 ml2. 2m Axis l m Let the axis pass through the geometric center of the rod. Find the moment of inertia by direct calculation. Ans: 3 4 ml2. Verify that your answers satisfy the parallel axis theorem.
6.! Some questions about the vector product C = A B. Prove the rules given in the notes for the magnitude and direction of C from the definition in terms of components. Let A and B lie in the x-y plane with A along the x-axis and B having direction with angle θ relative to A. One requires θ to be the angle between A and B that is no larger than π. Why? What are the rules for the vector products of the unit vectors (i,j,k)? Prove directly from the definition in terms of components that A B = B A. {Use the situation in (a).] 7.! Comment on the validity of these statements about forces and torques. If two forces obey F 1 + F 2 = 0, then they give zero torque about any point. If the total force on a body is zero and the total torque from those forces is zero about some point, then the total torque from those forces is zero about all points. Two forces acting along two separate parallel lines obey F 1 + F 2 = 0. There is a point between those lines about which the total torque is zero. If F 1 + F 2 = 0 and the forces both act along the same line, then they give zero total torque about any point. 8.! An object with a circular cross section of radius R and moment of inertia about its symmetry axis I = βmr 2 is rolling down an incline of height h and angle θ. ( µ s is large enough so that it rolls without slipping.) What is its speed at the bottom? Ans: v 2 = 2gh/(1 + β). θ What is the maximum value of θ so that it rolls without slipping? Ans: tanθ µ s β + 1. [Use F = ma, τ = Iα, and the rolling condition.] β
9.! The yo-yo shown in two views is resting on a floor that has a large enough value of µ s for the yo-yo to roll without slipping. The string is pulled at the angle shown with tension T. The yo-yo has mass m and moment of inertia I about its symmetry axis. The radius of the axle is r and that of the outer rim is R. θ If θ = 0 what is the acceleration (magnitude and direction) of the CM? 1 r /R [What is the direction of the friction force?] Ans: a = T, to right. 2 m + I /R r /R Repeat if θ = π /2. Ans: a = T, to left. 2 m + I /R For what value of θ is the acceleration zero? Ans: cosθ = r /R. 10.! Two questions about rolling. A ball of mass m, radius R, and moment of inertia 2 5 mr2, rolls without slipping at speed v on a horizontal floor. It starts up a frictionless incline. How high does it go before starting back down? Ans: h = v 2 /2g. The same ball rolls without slipping on the track shown, moving vertically at the end, at distance h below the level at which it started from rest. How high does it go in the air before falling? Ans: h = 5 7 h. h! [If there is no friction force, the angular speed of the balls s rotation cannot change.]
11.! The same ball is now released from rest on the track shown. It rolls without slipping around the inside of the circular portion which has radius d >> R. Find the minimum height h that will allow the ball to roll across the top of the circle. Ans: h = 2.7d. h 12.! The door shown has two hinges supporting it. Over time, what happens? The screws holding the bottom hinge to the door and the wall tend to work loose, because the door pulls on the hinge. The screws holding the top hinge to the door and the wall tend to work loose, because the door pulls on the hinge. Both sets of screws tend to pull loose, because the door pulls on both hinges. There is no force pulling either set of screws loose.! [Consider torques about the bottom hinge when the door is at rest.]