Two masses m and 2m are attached with each other by a rope passing over a frictionless and massless pulley. If the pulley is accelerated upwards with an acceleration ‘a’, what is the value of T?

If the pulley is accelerated at rate a, it is the same as if the weights became m(g +a) and 2m(g+a), in a coordinate system moving with the pulley, with gravitational accleration g + a.

There would be additional acceleration of the weights at a rate
a' = (g+a)(m/3m) = (g+a)/3
(2m would accelerate at rate a' down and m would accelerate up)

For the cable tension, let us apply Newton's second law to 2m.

2m*(g+a)-T = 2m*a' = 2m*(g+a)/3

T = 2m*(g+a)(2/3)= (4m/3)(g+a)

To find the value of T, the tension in the rope, we can apply Newton's second law of motion to the masses m and 2m separately.

For mass m:
The force acting on mass m is its weight (mg) and the tension in the rope (T), both acting downwards. The acceleration of mass m is the same as the pulley's acceleration (a), but in the opposite direction. So, applying Newton's second law, we have:
mg - T = ma ------(1)

For mass 2m:
The force acting on mass 2m is its weight (2mg) and the tension in the rope (T), both acting upwards. The acceleration of mass 2m is also the same as the pulley's acceleration (a), but in the same direction. So, applying Newton's second law, we have:
2mg + T = 2ma ------(2)

Now we have two equations (equation 1 and equation 2) with two unknowns (T and a). We can solve these equations simultaneously to find the values of T and a.

First, let's solve equation 1 for T:
T = mg - ma

Now substitute this into equation 2:
2mg + mg - ma = 2ma

Simplify:
3mg - ma = 2ma

Rearrange the terms:
3mg = 3ma

Divide both sides by 3m:
g = a

Now we know that the acceleration of the pulley is equal to the acceleration due to gravity.

Substitute this value of a back into equation 1 to solve for T:
T = mg - ma
T = mg - mg
T = 0

Therefore, when the pulley is accelerated upwards with an acceleration 'a', the tension in the rope (T) is equal to zero.

To find the value of T, we need to consider the forces acting on both masses and the tension in the rope.

Let's assume that the mass m is on the left side and the mass 2m is on the right side.

For the mass m:

1. The force acting downward is the weight, given by mg, where g is the acceleration due to gravity.
2. The force acting upward is the tension T in the rope.

For the mass 2m:

1. The force acting upward is the tension T in the rope.
2. The force acting downward is the weight, given by 2mg.

Since the two masses are connected by a rope passing over a pulley, the tensions on both sides have the same magnitude.

Now, let's consider the acceleration of the system:

1. The net force acting on the mass m is T - mg, in the upward direction.
2. The net force acting on the mass 2m is 2mg - T, in the downward direction.

According to Newton's second law, F = ma, where F is the net force acting on an object, m is the mass of the object, and a is the acceleration.

Applying this to both masses, we have:

1. For the mass m: T - mg = m * a
2. For the mass 2m: 2mg - T = 2m * a

We have two equations with two unknowns (T and a). We can now solve these equations simultaneously to find the value of T.

First, let's simplify the equations:

1. T - mg = ma
2. T = 2ma + 2mg

Now, let's solve for T:

Substituting equation 1 into equation 2,

T = 2ma + 2mg
T = 2(ma + mg)

We know that a = -a (since the acceleration of the pulley is upwards while the mass m is downwards).

T = 2(-ma - mg)
T = -2(ma + mg)

Thus, the value of T is -2(ma + mg).