Particles of mass 3m and 5m hang one at each end of a light inextensible string which passes over a pulley.the system is released from rest with the hanging parts taut and vertical.during the subsequent motion the resultant force exerted by the string on the pulley is of magnitude

particles of mass 3m and 5m hang one at each end of a light inextensible string which passes over a pulley the system is released from rest with the Hanging parts taut and vertical. During the subsequent motion the resultant force exerted by the string on the pulley is of magnitude

To find the magnitude of the resultant force exerted by the string on the pulley, we need to consider the forces acting on each particle.

Let's call the particle with mass 3m as Particle A and the particle with mass 5m as Particle B. Since the system is released from rest, we know that both particles will have an initial acceleration of zero.

For Particle A:
- There are two forces acting on it - the tension in the string pulling it upwards and the force due to gravity pulling it downwards.
- The magnitude of the force due to gravity is given by F_gravity_A = m_A * g, where m_A is the mass of Particle A and g is the acceleration due to gravity.
- Since Particle A has a mass of 3m, the magnitude of the force due to gravity is F_gravity_A = 3m * g.

For Particle B:
- There are also two forces acting on it - the tension in the string pulling it downwards and the force due to gravity pulling it upwards.
- The magnitude of the force due to gravity is given by F_gravity_B = m_B * g, where m_B is the mass of Particle B and g is the acceleration due to gravity.
- Since Particle B has a mass of 5m, the magnitude of the force due to gravity is F_gravity_B = 5m * g.

Since the particles are connected by a string and the string is inextensible, the tension in the string will be the same for both particles. Let's call this tension T.

Now, considering the motion of the system, we can see that the net force on Particle A is upward and the net force on Particle B is downward. This means that the tension in the string is pulling Particle A up and Particle B down.

By applying Newton's second law, we can equate the net forces on each particle to their respective masses times accelerations:

For Particle A:
T - F_gravity_A = m_A * a

For Particle B:
T - F_gravity_B = m_B * a

Since both particles have the same acceleration (as they are connected by the string), we can set the equations equal to each other:

T - F_gravity_A = T - F_gravity_B

Canceling out the tensions and rearranging the equation, we find:

F_gravity_B - F_gravity_A = 2m * g

Since the magnitude of the resultant force exerted by the string on the pulley is equal to the difference in gravitational forces on the particles, we have:

Magnitude of resultant force = |F_gravity_B - F_gravity_A|
= |5m * g - 3m * g|
= |2m * g|

So, the magnitude of the resultant force exerted by the string on the pulley is 2m * g.