Question

This is supposed to be a bit more challenging. Think carefully, take it step-by-step, see how far you can get on your own, then check with classmates, consult with me, TAs.... as necessary (11) Deluxe Atwood's machine. Consider the arrangement of two pulleys shown in the figure. Pulley 1 (on the left) is fastened to the ceiling. But pulley 2 (with mass 2 hanging from it) is free to move up and down. The string and the pulleys themselves are massless. The masses are not! Assume the pulleys are free to rotate without friction. a. For a given m2, what should be the choice of m1 such that the system is in equilibrium (balanced, not accelerating)? [hint: m1=m2 is NOT correct!!] b. The key to this problem is seeing that the motion of m1 and m2 are NOT identical, but also NOT entirely independent of each other. We saw a "constraint" problem like this on a previous problem set (go back and check out PS#1, prob. 9). The length of the string cannot change! State clearly how much (and in what direction) mass m2 moves when m1 moves by a given distance x. What does that say about the relative velocity and acceleration of the masses? c. Now consider the case when we are not in equilibrium. Find a general expression for the acceleration of m1 in terms of the masses. Check that your result for (a) agrees with this formula. If you want to maximize the magnitude of the acceleration of m1, should you pick m1 >> m2 or m1 << m2? d. Find an expression for the tension in the string in terms of the masses. For the equilibrium case (a), what is the tension? Does this make sense? e. For the case of maximum acceleration, what is the tension? Does this make sense in terms of the subsequent motion of m1?

This is supposed to be a bit more challenging. Think carefully, take it step-by-step, see how far you can get on your own, then check with classmates, consult with me, TAs.... as necessary

(11) Deluxe Atwood's machine.
Consider the arrangement of two pulleys shown in the figure. Pulley 1 (on the left) is fastened to the ceiling. But pulley 2 (with mass 2 hanging from it) is free to move up and down. The string and the pulleys themselves are massless. The masses are not! Assume the pulleys are free to rotate without friction.

a. For a given m2, what should be the choice of m1 such that the system is in equilibrium (balanced, not accelerating)? [hint: m1=m2 is NOT correct!!]
b. The key to this problem is seeing that the motion of m1 and m2 are NOT identical, but also NOT entirely independent of each other. We saw a "constraint" problem like this on a previous problem set (go back and check out PS#1, prob. 9). The length of the string cannot change! State clearly how much (and in what direction) mass m2 moves when m1 moves by a given distance x. What does that say about the relative velocity and acceleration of the masses?
c. Now consider the case when we are not in equilibrium. Find a general expression for the acceleration of m1 in terms of the masses. Check that your result for (a) agrees with this formula. If you want to maximize the magnitude of the acceleration of m1, should you pick m1 >> m2 or m1 << m2?
d. Find an expression for the tension in the string in terms of the masses. For the equilibrium case (a), what is the tension? Does this make sense?
e. For the case of maximum acceleration, what is the tension? Does this make sense in terms of the subsequent motion of m1?

This is supposed to be a bit more challenging. Think carefully, take it step-by-step, see how far you can get on your own, then check with classmates, consult with me, TAs.... as necessary

(11) Deluxe Atwood's machine.
Consider the arrangement of two pulleys shown in the figure. Pulley 1 (on the left) is fastened to the ceiling. But pulley 2 (with mass 2 hanging from it) is free to move up and down. The string and the pulleys themselves are massless. The masses are not! Assume the pulleys are free to rotate without friction.

a. For a given m2, what should be the choice of m1 such that the system is in equilibrium (balanced, not accelerating)? [hint: m1=m2 is NOT correct!!]
b. The key to this problem is seeing that the motion of m1 and m2 are NOT identical, but also NOT entirely independent of each other. We saw a "constraint" problem like this on a previous problem set (go back and check out PS#1, prob. 9). The length of the string cannot change! State clearly how much (and in what direction) mass m2 moves when m1 moves by a given distance x. What does that say about the relative velocity and acceleration of the masses?
c. Now consider the case when we are not in equilibrium. Find a general expression for the acceleration of m1 in terms of the masses. Check that your result for (a) agrees with this formula. If you want to maximize the magnitude of the acceleration of m1, should you pick m1 >> m2 or m1 << m2?
d. Find an expression for the tension in the string in terms of the masses. For the equilibrium case (a), what is the tension? Does this make sense?
e. For the case of maximum acceleration, what is the tension? Does this make sense in terms of the subsequent motion of m1?

Added by Jennifer M.

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