PHYS 211 Recitation Review Problems: Solutions

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Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions 9) Towing a Sled A mother tows her daughter on a sled on level ice. The friction between the sled and the ice is negligible, and the tow rope makes an angle of to the horizontal. The combined mass of the sled and the child is M . The sled has an acceleration in the horizontal direction of magnitude a. A) What is the tension T in the rope? We draw the free body diagram and acceleration diagram from the sled and note that only the horizontal component of the tension is responsible for the acceleration, allowing us to solve for the acceleration: a T cos Ma T Ma cos B) What is the magnitude of the normal force N exerted by the ice on the sled? The normal force we get from the vertical component: N T sin Mg N Mg T sin Mg Ma tan N Now the mother starts up an incline (angle ). She continues to pull at the same relative angle and at the same acceleration a. What new tension and normal force must exist? Work in a tilted coordinate system where the acceleration vector is along the “x-axis” and the normal to the plane it the “y-axis.” Then we find: a Ma Mg sin T cos Mg sin Ma T cos N T sin Mg cos N Mg cos T sin Mg cos Ma Mg sin tan N Check: These reduce to our previous expressions in the limit that goes to zero. Page 6 of 7
Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions 10) Blocks A block of mass m 1 is sitting on top of a more massive block of mass m 2 that rests on a horizontal table. There is a horizontal force of magnitude F pulling the top block to the right. There is friction between the two blocks and between the bottom block and the table, both with kinetic coefficient of friction μ . The blocks are connected by a massless, ideal rope that goes around a fixed massless and frictionless pulley. A) Draw FBDs and acceleration diagrams for both blocks 1 & 2 1 2 There are two action-reaction pairs – the friction and normal forces between the blocks F f . 12 and 12 B) What constraint relationship is there between the acceleration of block 1 and block 2 given the coordinate system(s) that you defined in your acceleration diagrams? a a We define the coordinate systems of the two blocks backward from each other so that a 1 = a 2 = a. If we chose to make “to the right” positive then the accelerations would have opposite signs. C) From your FBDs & acceleration diagrams write four equations of motion using Newton’s 2 nd law F F T f m a F : F F m g x,1 12 1 y,1 12 G,1 1 F T f f m a F : N F F m m g x,2 2G 12 2 y,2 12 G2 1 2 D) How many unknowns appear in your equations? Are any of your equations irrelevant, i.e. contain unknown(s) that appear nowhere else and whose value(s) you don’t care about? If this number is bigger than 4, do you have any other equations relating these unknowns? We have six unknowns: T, f12, f2G, F12 , N, a None of the equations are irrelevant since we need the normal forces to think about friction. We have two other equations relating friction and normal forces E) What are the frictional forces on the blocks (in terms of known quantities)? ; f F m g 12 12 1 f N m m g 2G 1 2 F) Solve for the acceleration a of the system. Note that to eliminate the tension T you might just add (or subtract) two of your equations from part C. We can add the two horizontal equations to cancel the tension: F T f T f f m a m a m m a 12 2G 12 1 2 1 2 1 1 1 2 a F 2f12 f2G F 2m1g m1 m2 g a m1 m2 m1 m2 m1 m2 F g 3m m Page 7 of 7
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PHYS 211 Recitation Review Problems: Solutions

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