A clerk can lift containers a vertical distance of 1 m, or roll them up a 2 m-long ramp to the same elevation. With the ramp, the applied force needed is.

Steps to Solve

1. Understanding Work Against Gravity When lifting an object vertically, the force needed is equal to the weight of the object, which can be expressed as $F = mg$, where $m$ is mass and $g$ is acceleration due to gravity.

2. Calculating Work Done in Vertical Lift The work done to lift an object a height $h$ is given by the formula: $$ W_{\text{lift}} = F \times h = mg \times h $$ In this case, $h = 1 \text{ m}$.

3. Using a Ramp to Lift the Same Height When the object is rolled up a ramp, the incline allows the force required to be less than the full weight. The height remains the same, but we apply a different force over a longer distance along the ramp.

4. Geometry of the Ramp If the ramp's length is $L = 2 \text{ m}$, and considering the same height, we can find the angle of the ramp, which can be calculated using: $$ \sin(\theta) = \frac{h}{L} = \frac{1}{2} $$ From this, the ramp angle $\theta$ can be determined.

5. Calculating Force on the Ramp The component of the gravitational force acting down the ramp can be calculated. The force required to push the container up the ramp is: $$ F_{\text{ramp}} = mg \sin(\theta) $$

Given that $\sin(\theta) = \frac{1}{2}$, we can find: $$ F_{\text{ramp}} = mg \cdot \frac{1}{2} = \frac{1}{2} mg $$

6. Comparison of Forces Finally, comparing the forces:

• Lifting vertically: $F_{\text{lift}} = mg$
• Rolling up the ramp: $F_{\text{ramp}} = \frac{1}{2} mg$

This shows that the force needed with the ramp is half the force needed to lift the container vertically.