Three blocks a,b, and c are connected by a light string and inextensible cord. Blocks a and b are placed on a smooth inclined plane which makes an angle of 30degree with the horizontal. Block c is suspended at the other end of the cord through a frictionless pulley. If masses of blocks a,b and c are 0.2kg, 0.1kg and 0.3 kg respectively, calculate the acceleration of the blocks.
component of weight down slope = (.1 + .2)g sin 30 = .15 g
F up slope
F - .15 g = .3 a rising blocks
.3 g - F = .3 a falling block
--------------------
.15 g = .6 a
a = .25 g = .25 * 9.81 = 2.45 m/s^2
To calculate the acceleration of the blocks, we need to apply Newton's second law of motion to each block separately.
Let's start with block a. Since block a is on a smooth inclined plane, the gravitational force acting on it can be resolved into two components: one perpendicular to the inclined plane and one parallel to the inclined plane.
The component of gravity perpendicular to the inclined plane does not contribute to the acceleration because it is balanced by the normal force exerted by the inclined plane. Therefore, we will focus on the component parallel to the inclined plane.
The parallel component of gravity can be calculated as follows:
Force_parallel_a = mass_a * acceleration_due_to_gravity * sin(angle)
where mass_a is the mass of block a, acceleration_due_to_gravity is the acceleration due to gravity (approximately 9.8 m/s^2), and angle is the inclination angle of the plane (30 degrees).
Next, let's consider block b. The only force acting on it is the tension in the string. The tension in the string can be calculated from the parallel component of gravity acting on block a. Therefore:
Tension = Force_parallel_a
Finally, let's analyze block c. The net force acting on block c is the difference between the force of gravity acting on it and the tension in the string. We can calculate it as follows:
Net_force_c = mass_c * acceleration
where mass_c is the mass of block c, and acceleration is the common acceleration of all the blocks.
Now, we can equate the net force on block c with its mass times acceleration:
Net_force_c = mass_c * acceleration
Substituting the values, we have:
mass_c * acceleration = mass_c * acceleration_due_to_gravity - Tension
Now we can substitute the value of tension from the previous step:
mass_c * acceleration = mass_c * acceleration_due_to_gravity - Force_parallel_a
Solving this equation will give us the acceleration of the blocks.