8. Two blocks of masses 50 Kg & 30 Kg, lie on two smooth inclined planes of 30° and 60° through a light rope which passes over a smooth pulley at junction of two planes. Calculate common acceleration and tension in rope
To calculate the common acceleration and tension in the rope, we can use Newton's second law of motion and the principles of mechanical equilibrium.
Let's start by analyzing the forces acting on each block individually.
For the 50 kg block:
1. There is the weight acting downward (mg) = (50 kg)(9.8 m/s^2) = 490 N.
2. There is a normal force acting perpendicular to the inclined plane.
3. There is a tension force in the rope acting upward.
For the 30 kg block:
1. There is the weight acting downward (mg) = (30 kg)(9.8 m/s^2) = 294 N.
2. There is a normal force acting perpendicular to the inclined plane.
3. There is a tension force in the rope acting downward.
Now, let's break these forces into components along the inclined plane for each block.
For the 50 kg block:
1. The weight has a component acting down the incline (490 N * sin(30°)).
2. The normal force has a component equal in magnitude, but opposite in direction, to the weight component along the incline (490 N * cos(30°)).
3. The tension force in the rope has a component acting up the incline.
For the 30 kg block:
1. The weight has a component acting down the incline (294 N * sin(60°)).
2. The normal force has a component equal in magnitude, but opposite in direction, to the weight component along the incline (294 N * cos(60°)).
3. The tension force in the rope has a component acting down the incline.
Since the two blocks are connected by a light rope passing over a smooth pulley, the magnitudes of the tension forces in the rope should be equal for both blocks.
Now, let's apply Newton's second law of motion to each block along the incline:
For the 50 kg block:
490 N * sin(30°) - T = (50 kg) * a
For the 30 kg block:
294 N * sin(60°) + T = (30 kg) * a
In the above equations, 'a' represents the common acceleration of both blocks, and 'T' represents the tension force in the rope.
To solve these equations simultaneously and find the values of 'a' and 'T', we need to substitute known values and solve the resulting system of equations.