Answer :
Answer:
To find the efficiency of the screw jack, we first need to determine the ideal mechanical advantage (IMA) and then use it to calculate efficiency.
Given:
- Pitch of the screw jack (p) = 0.5 cm
- Length of the arm (lever arm) of the screw jack (L) = 50 cm
- Mechanical advantage (MA) = 200
The ideal mechanical advantage (IMA) of the screw jack is calculated using the formula:
\[ \text{IMA} = \frac{\text{Length of travel}}{\text{Pitch}} \]
For one complete turn of the screw (which raises the jack by one pitch):
\[ \text{Length of travel} = 2 \pi \cdot \text{Lever arm} = 2 \pi \cdot 50 \text{ cm} = 100 \pi \text{ cm} \]
Therefore,
\[ \text{IMA} = \frac{100 \pi \text{ cm}}{0.5 \text{ cm}} = 200 \pi \]
Now, efficiency (η) is given by the formula:
\[ \eta = \frac{\text{MA}}{\text{IMA}} \times 100\% \]
Substitute the given values:
\[ \eta = \frac{200}{200 \pi} \times 100\% \]
\[ \eta = \frac{1}{\pi} \times 100\% \]
\[ \eta = \frac{100}{\pi}\% \]
Therefore, the efficiency of the screw jack is:
\[ \boxed{\frac{100}{\pi}\%} \]
This is the exact expression for the efficiency of the machine in terms of \( \pi \).
Explanation:
