Question

A cord is used to lower vertically a block of mass M, through a distance d at a constant downward acceleration of $\dfrac{g}{8}$. Then the work done by the cord on the block is-
A. $\dfrac{{Mgd}}{8}$
B. $\dfrac{{3Mgd}}{8}$
C. $Mgd$
D. $\dfrac{{ - 7Mgd}}{8}$

Answer 1

Hint- The block is hanging from a cord so the tension will be in the upwards direction along the cord. The force acting on it due to gravity will be mg and the downward acceleration is given in the question as g/8. So, the net force will be the difference between the gravitational force and the Tension. Refer to the solution below.

Complete step-by-step answer:
Formula used: $F = ma$, ${W_{CB}} = \vec F.\vec d$
Since the tension is being applied on the block in the upwards direction and the gravitational force on the block is in the downward direction, the net force will be-
$\Rightarrow {F_{net}} = Mg - T$
Now, according to the Newton’s second law of Motion, we can say that-
$\Rightarrow F = ma$
$\Rightarrow F = Ma$ (mass of the block is M)
Where m stands for mass of the body and a stand for the acceleration.
Equation the above formulas, we get-
$\Rightarrow Mg - T = Ma \\\ \\\ \Rightarrow T = Mg - Ma \\\$
Substituting the value of acceleration i.e. g/8 as per given in the question in the above equation-
$\Rightarrow T = Mg - M\dfrac{g}{8} \\\ \\\ \Rightarrow T = \dfrac{{7Mg}}{8} \\\$
Work done by the cord on the block-
$\Rightarrow {W_{CB}} = \vec F.\vec d$ (dot product)
In this case, force applied is in the upwards direction which is Tension. Since the tension in working in the upwards direction and the displacement is in the downwards direction, the angle between these two forces will be 180 degrees. So-
$\Rightarrow {W_{CB}} = \left| {\vec T} \right|\left| {\vec d} \right|\cos 180^\circ$
Substituting the values of tension in the above formula, we get-
$\Rightarrow {W_{CB}} = \dfrac{{7Mg}}{8}d\left( { - 1} \right)$ ($\cos 180^\circ = - 1$)
$\Rightarrow {W_{CB}} = - \dfrac{{7Mgd}}{8}$
Hence, option D is the correct option.

Note: Newton's second motion law is closely related to Newton's first motion law. This mathematically states the relation of cause and effect between force and motion changes. Newton's second law of motion is more abstract, and it is commonly used to measure what happens in circumstances involving a force.