Question

A uniform ladder 10.0 m long is leaning against a frictionless wall at an angle of 60.0° above the horizontal. What is the weight of the ladder?

Answer 1

To determine the weight of the ladder, we can start by analyzing the forces acting on it and using the concept of static equilibrium. When the ladder is in static equilibrium, the net force and the net torque acting on it must be zero.

1. **Identify Forces**:
- The weight (W) of the ladder acts vertically downward at its center of mass, which is located at its midpoint (5.0 m from either end).
- The normal force (N) from the wall acts horizontally on the top of the ladder.
- The normal force (N_ground) from the ground acts vertically upwards at the base of the ladder.
- The friction force at the ground is zero since the wall is frictionless.

2. **Setting Coordinates**:
- Let the bottom of the ladder be at the origin (0, 0).
- The wall contact will be at an angle of 60° above the horizontal.

3. **Taking moments about the base of the ladder**:
- To find W (the weight of the ladder), we can set the sum of moments about the bottom of the ladder to zero.

- The lever arm of the weight (W) about the base is \( \frac{10.0}{2} \cdot \cos(60°) \) since the weight acts downward at the midpoint of the ladder.

- The distance of the normal force (N) from the base is \( 10.0 \cdot \sin(60°) \).

4. **Torque Equation**:
\[
\text{Sum of moments about the base} = 0:
\]
\[
W \cdot \left(\frac{10.0}{2} \cdot \cos(60°)\right) - N \cdot (10.0 \cdot \sin(60°)) = 0
\]
\[
W \cdot 5.0 \cdot 0.5 - N \cdot 10.0 \cdot \frac{\sqrt{3}}{2} = 0
\]

Simplifying,
\[
\frac{W}{2} = N \cdot 5\sqrt{3}
\]
\[
W = 10N \sqrt{3}
\]

5. **Vertical Forces**:
The vertical forces are also in balance:
\[
N_{\text{ground}} - W = 0
\]
Thus,
\[
N_{\text{ground}} = W
\]

6. **Horizontal Forces**:
The horizontal forces should also balance, which can be expressed as:
\[
N = N_{\text{ground}}
\]

By substituting \(N\) into the torque equation, we can find an expression for \(W\) but we need more information to calculate a specific weight (like the ladder's material density or its mass).

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To sum up, to precisely find the weight of the ladder, we'd need additional information such as the density or mass of the material from which the ladder is made. In practical measurements, one would ensure to weigh the ladder directly using scales if such information isn't provided. Would you like further details on solving it or more context?