Three objects with masses m1, m2 , and m3, respectively, are attached by strings over frictionless pulleys. If the system is released from rest, find the acceleration of each mass and each tension if m3 moves down.
To find the acceleration of each mass and the tension in each string, we need to use Newton's second law of motion and apply it to each individual mass.
Let's start by analyzing each mass separately:
1. Mass m1: The tension in the string connected to m1 is pulling it to the right. The only force acting on m1 is the tension force, so we can write the equation for m1 as:
T1 - m1 * g = m1 * a --(1)
where T1 is the tension in the string connected to m1, g is the acceleration due to gravity (9.8 m/s^2), m1 is the mass of object m1, and a is the acceleration of the system.
2. Mass m2: The tension in the string connected to m2 is pulling it to the left. The gravitational force acting on m2 is m2 * g. The net force acting on m2 is the difference between the tension and the gravitational force, so we can write the equation for m2 as:
m2 * g - T2 = m2 * a --(2)
where T2 is the tension in the string connected to m2, m2 is the mass of object m2, and a is the acceleration of the system.
3. Mass m3: The tension in the string connected to m3 is also pulling it to the left. The gravitational force acting on m3 is m3 * g. The net force acting on m3 is the difference between the tension and the gravitational force, so we can write the equation for m3 as:
m3 * g - T3 = m3 * a --(3)
where T3 is the tension in the string connected to m3, m3 is the mass of object m3, and a is the acceleration of the system.
Now, we have three equations (equations 1, 2, and 3) with three unknowns (T1, T2, and a). We can solve these equations simultaneously to find the values of T1, T2, and a.
To find the acceleration of each mass, we can solve for a using any of the three equations since all the values of T1, T2, and T3 should be the same once the system reaches equilibrium. Let's use equation 1 and solve for a:
T1 - m1 * g = m1 * a
a = (T1 - m1 * g) / m1
To find the tension in each string, we can use equations 2 and 3 and solve for T2 and T3, respectively.
Once we know the values of T1, T2, and a, we can substitute them back into equations 2 and 3 to find the specific values for T2 and T3.