Question

two blocks with masses M1 and M2 hang one under the other. For this problem take the positive direction to be upward, and use g for the magnitude of the acceleration due to gravity

1-find T2 the tension in the lower rope
2 - Find T1 the tension in the upper rope;

For question 3 and 4 the blocks are now acceleration upward due to the tension in the strings with acceleration of magnitude a.
3- Find T2 the tension in the lower rope
4- Find T1 the tension in the upper rope

express answers in terms of M1, M2 and g

Answer 1

Concepts and reason

The main concepts required are the Newton’s second law.

Initially, find the tension in the rope on which mass is hanging vertically in both the conditions using the Newton’s second law. Then, in case of acceleration upward, the tension in the lower rope and the upper rope has been calculated using the Newton’s second law.

Finally, the tension in the ropes can be found by writing force equation for the masses.

Fundamentals

Consider a system of two masses ${M_1}$and ${M_2}$ hanging one under another through a rope.

The tension ${T_1}$ in the upper rope is,

${T_1} = \left( {{M_1} + {M_2}} \right)g$

Here g is the acceleration due to gravity.

The tension ${T_2}$ in the lower rope is,

${T_2} = {M_2}g$

Let the blocks be accelerated upward with acceleration a, in this case tension ${T_1}$ in the upper rope and tension ${T_2}$ in the lower rope is given as follows:

${T_1} = \left( {{M_1} + {M_2}} \right)\left( {g + a} \right)$

${T_2} = {M_2}\left( {g + a} \right)$

1)

Find the tension in the lower rope.

The gravitational pull of the earth attracts the body downwards and the tension in the rope acts upward.

So, the force equation of the mass ${M_2}$is given as follows:

$\begin{array}{c}\\{T_2} - {M_2}g = 0\\\\{T_2} = {M_2}g\\\end{array}$

(2)

Find the tension ${T_1}$in the upper rope.

The three forces acting on the mass ${M_1}$are the gravitational pull by the earth acting downwards, the upward pull due to tension in the upper rope and the downward pull due to tension in lower rope.

The force equation of mass ${M_1}$is given below:

$\begin{array}{c}\\{T_1} - {T_2} - {M_1}g = 0\\\\{T_1} = {T_2} + {M_1}g\\\end{array}$

Substitute $\left( {{M_2}g} \right)$for ${T_2}$ in the last expression as follows:

${T_1} = \left( {{M_1} + {M_2}} \right)g$

(3)

Find the tension ${T_2}$ in the lower rope.

The lower body accelerates under the net effect of two forces acting on it. The forces acting on the mass ${M_2}$are the gravitational pull by the earth acting downwards and the upward pull due to tension in the lower rope.

The force equation of mass ${M_1}$is given below:

${T_2} - {M_2}g = {M_2}a$

Solve it for tension as below:

${T_2} = {M_2}\left( {g + a} \right)$

(4)

Find the tension ${T_1}$ in the upper rope.

The upper body accelerates under the net effect of three forces acting on it. The forces acting on the mass ${M_1}$are the gravitational pull by the earth acting downwards, the downward pull by the lower rope and the upward pull due to tension in the upper rope.

The force equation of mass ${M_1}$is given below:

$\begin{array}{c}\\{T_1} - {T_2} - {M_1}g = {M_1}a\\\\{T_1} = {T_2} + {M_1}\left( {g + a} \right)\\\end{array}$

Substitute $\left\{ {{M_2}\left( {g + a} \right)} \right\}$ for tension in the lower rope as follow:

${T_1} = \left( {{M_1} + {M_2}} \right)\left( {g + a} \right)$

Ans: Part 1

The tension ${T_2}$ in the lower rope is $\left( {{M_2}g} \right)$.

Part 2

The tension ${T_1}$ in the lower rope is $\left( {{M_1} + {M_2}} \right)g$.

Part 3

The tension ${T_2}$ in the lower rope is $\left\{ {{M_2}\left( {g + a} \right)} \right\}$.

Part 4

The expression for tension ${T_1}$ in the upper rope is $\left( {{M_1} + {M_2}} \right)\left( {g + a} \right)$.