A freely hanging block of mass 5 kg is attached to a massless string which is wrapped around a pulley of mass 10 kg and radius 0.1 m. What is the acceleration of the block?
To find the acceleration of the block, we need to set up an equation using Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
First, let's consider the forces acting on the system. We have the weight of the block pulling it downward, which can be calculated as the mass of the block multiplied by the acceleration due to gravity (9.8 m/s^2). This force is equal to m1 * g, where m1 is the mass of the block.
Next, we have the tension in the string. Since the pulley is massless, the tension in the string is the same on both sides of the pulley. Therefore, the tension in the string is also acting on the block and can be written as T.
Finally, we have to consider the rotational dynamics of the pulley. The net torque acting on the pulley is equal to the moment of inertia of the pulley (I) multiplied by its angular acceleration (alpha). This net torque is due to the tension in the string and the force of friction between the axle of the pulley and its support. Since the pulley is not accelerating rotationally, the net torque is zero.
Using the equations above, we can write down the following equations:
Equation 1: m1 * g - T = m1 * a (from Newton's second law for the block)
Equation 2: T * r = I * alpha (from rotational dynamics of the pulley)
Now, let's solve for the acceleration (a):
Rearrange Equation 1 to isolate the acceleration:
m1 * g - T = m1 * a
=> a = (m1 * g - T) / m1
To find the tension (T), we need to substitute it from Equation 2:
T * r = I * alpha
The moment of inertia (I) for a solid cylinder is equal to (1/2) * m2 * r^2, where m2 is the mass of the pulley.
Substituting, we get:
T * r = (1/2) * m2 * r^2 * alpha
Simplifying, we have:
T = (1/2) * m2 * r * alpha
Now, substitute this into Equation 1 to find the acceleration:
a = (m1 * g - (1/2) * m2 * r * alpha) / m1
Since the pulley is not accelerating rotationally (alpha is zero), we can simplify the equation further:
a = (m1 * g - 0) / m1
=> a = g
Therefore, the acceleration of the block is equal to the acceleration due to gravity, which is approximately 9.8 m/s^2.