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Step-by-step Solution
Step 1: Understand the Problem
A hydraulic press lifts a 100 kg load when a certain mass m is placed on the smaller piston. We are asked to determine the load it can lift if the diameter of the larger piston is increased 4 times and the diameter of the smaller piston is decreased 4 times, while keeping the same mass m on the smaller piston.
Step 2: Apply Pascal’s Law
Pascal’s law states:
\frac{F_1}{A_1} = \frac{F_2}{A_2}
where F_1 and F_2 are the forces on the smaller and larger pistons respectively, and A_1 and A_2 are their respective areas.
Step 3: Express the Initial Configuration
In the initial setup, the force on the larger piston is due to the 100 kg load, and the force on the smaller piston is due to mass m . Thus,
\frac{100g}{A_1} = \frac{mg}{A_2} \quad \dots (i)
Here, g is the acceleration due to gravity, A_1 is the area of the smaller piston, and A_2 is the area of the larger piston.
Step 4: Express the Final Configuration
When the diameter of the larger piston is increased 4 times, its radius is also increased 4 times. Therefore, its area increases by a factor of 4^2 = 16 . Conversely, when the diameter of the smaller piston is decreased 4 times, its area is reduced by a factor of 16.
If M is the new load that can be lifted:
\frac{Mg}{16A_1} = \frac{mg}{\bigl(\frac{A_2}{16}\bigr)} \quad \dots (ii)
Step 5: Divide Equation (i) by Equation (ii)
Dividing (i) by (ii):
\frac{\frac{100g}{A_1}}{\frac{Mg}{16 A_1}} \; \bigg/ \; \frac{\frac{mg}{A_2}}{\frac{mg}{\left(\frac{A_2}{16}\right)}} = 1
which simplifies to
\frac{100 \times 16}{M} = \frac{1}{16}.
Solving for M gives
M = 100 \times 16 \times 16 = 25600 \, \text{kg.}
Step 6: Conclusion
The hydraulic press can lift 25600 kg under the new conditions.
