The main results are:
* The acceleration of the system is 10.72 m/s^2.
* The tension at the top of the rope is 160.8 N.
* The tension at the midpoint of the rope is 160.8 N.
a) Free body diagrams:
Block 1:
* Upward force: 200 N
* Weight: 6.0 kg * 9.8 m/s^2 = 58.8 N
* Tension in rope (T1): pointing downwards
Block 2:
* Weight: 5.0 kg * 9.8 m/s^2 = 49.0 N
* Tension in rope (T2): pointing upwards
Rope:
* Weight: 4.0 kg * 9.8 m/s^2 = 39.2 N
* Tension at top (T1)
* Tension at midpoint
* Tension at bottom (T2)
b) Acceleration of the system:
To determine the acceleration of the system, we need to consider the net force acting on the system. The net force is equal to the sum of all forces acting on the system. In this case, the net force is equal to the upward force (200 N) minus the weight of the rope (39.2 N).
Net force = 200 N - 39.2 N = 160.8 N
We can now use Newton's second law to relate the net force to the acceleration of the system:
Net force = mass * acceleration
where mass is the total mass of the system (6.0 kg + 5.0 kg + 4.0 kg = 15.0 kg).
Plugging in the values, we get:
160.8 N = 15.0 kg * acceleration
acceleration = 10.72 m/s^2
Therefore, the acceleration of the system is 10.72 m/s^2.
c) Tension at the top of the rope:
The tension at the top of the rope (T1) is equal to the force applied to the system (200 N) minus the weight of the rope (39.2 N).
T1 = 200 N - 39.2 N = 160.8 N
Therefore, the tension at the top of the rope is 160.8 N.
d) Tension at the midpoint of the rope:
The tension at the midpoint of the rope is equal to the average of the tension at the top (T1) and the tension at the bottom (T2).
Tension at midpoint = (T1 + T2) / 2
Since the rope is uniform, the tension at the top (T1) is equal to the tension at the bottom (T2). Therefore, the tension at the midpoint is equal to the tension at either end of the rope.
Tension at midpoint = T1 = T2 = 160.8 N
Therefore, the tension at the midpoint of the rope is 160.8 N.