Question

An open knife edge of mass ‘m’ is dropped from a height ‘h’ on a wooden floor. If the knife penetrates up to depth ‘d’ into the wood, the average resistance offered by the wood to the knife edge is:
A. \[mg\]
B. \[mg\left( 1-\dfrac{h}{d} \right)\]
C. \[mg\left( 1+\dfrac{h}{d} \right)\]
D. \[mg{{\left( 1+\dfrac{h}{d} \right)}^{2}}\]

Answer 1

Hint: In this question we have been asked to calculate the resistance offered by the wood to the knife-edge that penetrated up to the depth d in the wood when it was dropped from a height h. To solve this question, we shall first calculate the work done by the wood to stop the knife. We know that the work done on the knife will be due to the potential energy of the knife stored at height h.

Complete step-by-step solution
The knife falling from height h penetrates the wood up to the depth d as shown in the figure below.

Now, from the figure, we can say that the net energy, say ‘E’ lost during the whole process will be due to the potential energy of knife falling from height h to depth d,
Therefore, we can say that,
\[E=mg(h+d)\] ……………. (1)
We know that the work done to stop the knife-edge at depth d will be given as the product of resistance force, say ‘F’ applied by the wood and the depth d.
Therefore,
\[W=F\times d\]………….. (2)
Now, we know that the work done will be due to the potential energy of the knife
Therefore, we can say that,
\[W=E\] …………… (3)
Now, from (1), (2) and (3) we can say that,
\[F\times d=mg(h+d)\]
On solving,
\[F=\dfrac{mg(h+d)}{d}\]
Therefore,
\[F=mg\left( 1+\dfrac{h}{d} \right)\]
Therefore, the correct answer is option B.

Note: The energy that is stored in an object is known as potential energy. The potential energy is stored in an object due to the displacement of the object from the datum plane. An example of potential energy is the stored water in dams. The water in dams is stored at a certain height. Here the ground is considered as a datum and the displacement is vertical. This water now has the potential to do work.