2 masses m1 and m2 are connected at ends of in-extensible string that passes over pulley find acceleration and tension in pulley.
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To find the acceleration and tension in the pulley, we can use Newton's second law of motion.
First, we need to set up the coordinate system. Let's assume that m1 is hanging on one side of the pulley and m2 is on the other side, and they are connected by an inextensible string. We'll also assume positive acceleration when m1 moves downward and positive tension when to the right.
Now, let's break down the forces acting on the masses:
For m1:
- The force of gravity acting downward with a magnitude of m1 * g, where g is the acceleration due to gravity.
- The tension in the string acting upward with an unknown magnitude.
For m2:
- The force of gravity acting downward with a magnitude of m2 * g.
- The tension in the string acting downward with an unknown magnitude.
The acceleration of the masses is the same since they are connected by the same string. Let's assume the acceleration is "a". According to Newton's second law, we have the following equations:
For m1:
m1 * g - T = m1 * a
For m2:
T - m2 * g = m2 * a
Next, we consider the pulley. As the masses move, the pulley rotates, and there is a torque acting on it due to the tension in the string. The tension on both sides of the pulley creates a net torque of zero because the string is inextensible. If we consider the pulley to be a disk with a moment of inertia "I", the torque acting on the pulley is given by:
Torque = T * R
Here, "R" is the radius of the pulley.
Assuming the pulley's mass is negligible, we can apply Newton's second law for rotational motion:
Torque = I * α
Since the pulley's moment of inertia is given by I = 0.5 * M * R^2, where M is the mass of the pulley, and the angular acceleration (α) is related to the linear acceleration (a) by α = a / R, we can rewrite the equation as:
T * R = 0.5 * M * R^2 * (a / R)
Canceling out "R" and rearranging the equation, we get:
T = 0.5 * M * a
Now, we have two equations with two unknowns (T and a). We can solve this system of equations simultaneously.
Solving the system of equations, the values for acceleration and tension can be obtained.