Two masses are hooked together over a pulley and can slide without friction. If M1 = 6 kg, theta 1 = 45 degrees, and theta 2 = 17 degrees, what must M2 be for the blocks to move with constant velocity?
To determine what M2 must be for the blocks to move with constant velocity, we need to analyze the forces acting on the system.
Let's start by drawing a free-body diagram for each mass.
For M1:
- The weight force (mg) is acting vertically downward.
- The tension in the string is acting horizontally and to the right.
- The normal force (N) is acting vertically upward, but since there is no vertical acceleration, N is equal to mg.
For M2:
- The weight force (mg) is acting vertically downward.
- The tension in the string is acting horizontally and to the left.
- The normal force (N) is acting vertically upward, but since there is no vertical acceleration, N is equal to mg.
Since the blocks are connected by a string without friction, the magnitudes of the tensions in the string on both sides of the pulley are equal. Therefore, the tension in the string pulling M1 to the right is the same as the tension pulling M2 to the left.
Now, let's apply Newton's second law to each mass.
For M1:
- In the horizontal direction, the tension is the only force acting, and the acceleration is zero (constant velocity). Therefore, T = M1 * a, where a = 0.
For M2:
- In the horizontal direction, the tension is the only force acting, and the acceleration is zero (constant velocity). Therefore, T = M2 * a, where a = 0.
Since the tensions on both sides of the pulley are equal, we can equate the two expressions for tension:
M1 * 0 = M2 * 0
Since any number multiplied by zero is zero, this equation becomes:
0 = 0
This equation is true for any value of M2. Therefore, there is no restriction on the value of M2 for the blocks to move with constant velocity. In other words, M2 can be any mass, and the blocks will still move with constant velocity as long as there is no friction and the angles remain constant.