Question

Vicky opens a nut using a $45\text{cm}$ long wrench by applying a force of $220\text{N}$.

• If the length of the wrench used had been $0.3\text{m}$ instead, would it have been easier for him to open the nut?
• Prove your statement by finding the effort needed in case of using the shorter wrench.

Answer 1

Given:
The length of the wrench in the first case, ${\text{l}}_1=45\text{cm}$
The force/effort applied in the first case, ${\text{F}}_1=220\text{N}$

The moment of force generated in this turning effect of opening the nut using a wrench will be the product of the force applied and the length of the wrench used.

i.e. in the first case, the moment of force, ${\tau }_1={\text{F}}_1\times {\text{l}}_1$
$=220\text{N}\times 45\text{cm}$
$=9900\text{N-cm}$ or $99\text{N-m}------\left(1\right)$

For the same moment of force, if the length of the wrench is reduced, the force/effort needed to open the nut, will be more.

So, when a wrench of length $0.3\text{m}$ is used instead, the new force needed will be:

$99\text{N-m}={\text{F}}_{new}\times 0.3\text{m}$

$⟹{\text{F}}_{new}=\frac{99\text{N-m}}{0.3\text{m}}=\frac{990}{3}=330\text{N}$

Therefore, the effort needed in the shorter wrench case is more and is equal to $330\text{N}$.