A 2 kg block is resting on a 37 degree frictionless inclined plane, attached by a string to a frictionless pulley at the top of the plane, over which a second mass of 3.3 kg hangs down. Find the magnitude of the hanging mass’s acceleration.
To find the magnitude of the hanging mass's acceleration, we can use Newton's second law of motion and the principles of forces and acceleration.
1. First, let's draw a diagram to visualize the problem. We have a 2 kg block on an inclined plane at an angle of 37 degrees and a 3.3 kg mass hanging vertically on the other side.
2. Decompose the gravitational force acting on the 2 kg block into two components: one parallel to the inclined plane and one perpendicular to it.
3. The force perpendicular to the inclined plane is equal to the weight of the block, which is given by the formula:
Weight = mass * acceleration due to gravity
= 2 kg * 9.8 m/s^2
= 19.6 N
4. The force parallel to the inclined plane can be found using trigonometry. The formula for the force parallel to the inclined plane is:
Force parallel = weight * sin(theta)
= 19.6 N * sin(37 degrees)
= 11.863 N
5. Since the plane is frictionless, the only force acting on the block parallel to the plane is the tension in the string. Therefore, the acceleration of the block can be determined using Newton's second law:
Force parallel = mass * acceleration
11.863 N = 2 kg * acceleration
acceleration = 11.863 N / 2 kg
acceleration ≈ 5.932 m/s^2
6. Now, let's consider the 3.3 kg mass hanging vertically. The weight of this mass is:
Weight = mass * acceleration due to gravity
= 3.3 kg * 9.8 m/s^2
= 32.34 N
7. The tension in the string is equal to the force required to accelerate the mass:
Tension = mass * acceleration
= 3.3 kg * acceleration
≈ 3.3 kg * 5.932 m/s^2
≈ 19.5606 N
8. Since the tension in the string is the only force acting on the 3.3 kg mass, the acceleration of the hanging mass is given by:
acceleration = Tension / mass
= 19.5606 N / 3.3 kg
≈ 5.937 m/s^2
Therefore, the magnitude of the hanging mass's acceleration is approximately 5.937 m/s^2.