The radius of one arm of a hydraulic lift is three times the radius of the other arm. What force should be applied on the narrow arm so as to lift 50 kg at the wider arm?

Options

A. 60 N
B. 54.4 N
C. 26.7 N
D. 30 N

Answer

To solve the problem of determining the force required to lift a 50 kg weight using a hydraulic lift with arms of different radii, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Let the radius of the narrow arm be \( r \).
- The radius of the wider arm is \( 3r \).
- The mass to be lifted on the wider arm is \( 50 \, \text{kg} \).

2. **Calculate the Weight on the Wider Arm:**
- The weight \( W \) can be calculated using the formula:
\[ W = m \cdot g \]
- Where \( m = 50 \, \text{kg} \) and \( g \approx 9.8 \, \text{m/s}^2 \).
- Thus,
\[ W = 50 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 490 \, \text{N} \]

3. **Apply Pascal's Law:**
- According to Pascal's law, the pressure applied on both arms of the hydraulic lift is equal:
\[ P_1 = P_2 \]
- Where:
- \( P_1 = \frac{F}{A_1} \) (Pressure at the narrow arm)
- \( P_2 = \frac{W}{A_2} \) (Pressure at the wider arm)

4. **Calculate the Areas of the Arms:**
- The area \( A \) of a circle is given by \( A = \pi r^2 \).
- For the narrow arm:
\[ A_1 = \pi r^2 \]
- For the wider arm:
\[ A_2 = \pi (3r)^2 = 9\pi r^2 \]

5. **Set Up the Pressure Equation:**
- Substitute the areas into the pressure equations:
\[ \frac{F}{\pi r^2} = \frac{490 \, \text{N}}{9\pi r^2} \]

6. **Simplify the Equation:**
- Cancel \( \pi r^2 \) from both sides:
\[ F = \frac{490 \, \text{N}}{9} \]

7. **Calculate the Force \( F \):**
- Now compute \( F \):
\[ F = \frac{490}{9} \approx 54.44 \, \text{N} \]

### Final Answer:
The force that should be applied on the narrow arm to lift the 50 kg weight at the wider arm is approximately **54.4 N**.

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