P121. Connected Object Problem

Don't attempt this problem until you've studied the Guide for Solving Connected Object Problems and the audio-video tutorials.

Use of subscripts, organization, and clarity of labels are even more important in this problems than in others that you've done.

Refer to the diagram to the right. A force T1 pulls horizontally on a block of mass M1 that slides on a rough, horizontal surface. The block is connected by a horizontal string to a second block of mass M2 on the same surface. The tension in the string connecting the blocks is T2. The coefficient of kinetic friction between either block and the surface is μk. Assume the strings have negligible mass. Do the following.

    1. Draw a picture of the situation and show the direction of the acceleration. Define the directions of the coordinate axes.

    2. In order to be able to say that both blocks have the same horizontal acceleration, what must be true of the strings?

    3. Draw a force diagram for each block. Distinguish forces of the same type with appropriate subscripts for the two blocks.

    4. Tell how you used Newton's 3rd Law in part c.

    5. Write the two net force equations for each block. That's a total of 4 equations. (As an example of the kind of subscripting you need to do, a properly subscripted net force for block 2 in the x direction is Fnet,x2.)

    6. Using the 4 equations above as well as Newton's 2nd Law and the equation that relates kinetic friction force to normal force, solve algebraically for the acceleration of the system, ax. Give your final equation for axin simplest algebraic form in terms of M1, M2, T1, μk, and g only. (T2 may not appear in the equation.)

    7. You can check your equation for ax by considering the two blocks and the string as a system. What are i) the net force on system and ii) the mass of the system? Do the two methods of finding the acceleration agree?

    8. Now determine an equation in simplest form for T2 in terms of M1, M2, and T1 only. You can do this by starting with an Fnet equation that contains T2. Substitute the expression that you found for the acceleration and simplify. Interestingly, the tension in the string doesn't depend on the coefficient of kinetic friction.

    9. Check your equation for T2 as follows.

      1. If there is no second block, that is M2 = 0, what does your equation give for T2? Does this make sense and why?

      2. If there is no first block, that is M1 = 0, what does your equation give for T2? Does this make sense and why?

      3. If the two blocks have equal mass, how do the two tension forces relate?

 


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