Two blocks are connected by a rope that passes over a set of pulleys. One block has a weight of 412 N, and the other has a weight of 908 N. The rope and the pulleys are massless and there is no friction. (a) What is the acceleration of the lighter block? (b) Suppose that the heavier block is removed, and a downward force of 908 N is provided by someone pulling on the rope, as part (b) of the drawing shows. Find the acceleration of the remaining block.
To solve this problem, we can apply Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
(a) To find the acceleration of the lighter block when both blocks are connected, we need to consider the net force acting on the system. Since the rope and pulleys are massless and there is no friction, the tension force in the rope will be the same throughout the system. Let's denote the tension force as T.
For the lighter block:
Using Newton's second law, we have:
T - Weight of the lighter block = Mass of the lighter block * Acceleration
Substituting the given values:
T - 412 N = Mass of the lighter block * Acceleration
For the heavier block:
Using Newton's second law, we have:
Weight of the heavier block - T = Mass of the heavier block * Acceleration
Substituting the given values:
908 N - T = Mass of the heavier block * Acceleration
Since T is the same in both equations, we can set them equal to each other and solve for acceleration:
T - 412 = 908 - T
Simplifying the equation:
2T = 908 + 412
2T = 1320
T = 660 N
Now we can substitute the value of T back into either of the previous equations to solve for the acceleration of the lighter block:
660 - 412 = Mass of the lighter block * Acceleration
248 = Mass of the lighter block * Acceleration
We don't have the value of the mass of the lighter block, so we can't find the numerical value of the acceleration. However, we know that the acceleration will be 248 divided by the mass of the lighter block.
(b) In this case, the heavier block is removed, and the net force on the remaining block is provided by the person pulling on the rope, with a downward force of 908 N. Let's denote the acceleration of the remaining block as a'.
Using Newton's second law, we have:
Force on the remaining block = Mass of the remaining block * Acceleration of the remaining block
Substituting the given values:
908 N = Mass of the remaining block * a'
Since the mass of the remaining block is not given, we cannot find the numerical value of the acceleration. The acceleration will be 908 divided by the mass of the remaining block.