Question

A section board is 8 ft long. How high will one end be raised so that a block on it will just begin to slide down if the coefficient of friction is 0.3?

Answer 1

To determine how high one end of the section board needs to be raised for a block on it to just begin sliding down, we can analyze the forces involved. The critical point is when the force due to gravity acting downhill is equal to the force due to friction uphill. We can calculate the height required using the following steps:

1. Convert the length of the section board from feet to meters. Since 1 ft is approximately 0.3048 meters, the length of the section board is 8 ft × 0.3048 m/ft = 2.4384 m.

2. Determine the normal force acting on the block. The normal force is equal to the weight of the block, which is the product of its mass (m) and the acceleration due to gravity (g). The value of g is approximately 9.8 m/s².

3. Calculate the force of friction using the coefficient of friction (μ) and the normal force. The force of friction (F_friction) is given by the equation F_friction = μ × N, where N is the normal force.

4. Set up an equilibrium equation. At the critical point when the block is about to slide, the downhill force (due to gravity) is equal to the uphill force (due to friction). Therefore, we have F_gravity = F_friction.

5. Calculate the height (h) required for the block to begin sliding. The height can be determined using the equation F_gravity = m × g = F_friction = μ × N. Since N is the product of the normal force and the cosine of the angle of inclination, we can write m × g = μ × (m × g × cosθ), where θ is the angle of inclination. Solving for h, we get h = tanθ.

Let's perform the calculations:

1. The length of the section board is 2.4384 m.

2. The normal force N is equal to m × g. Since the mass of the block is not provided, we can assume a value, such as 1 kg. Therefore, N = 1 kg × 9.8 m/s² = 9.8 N.

3. The force of friction F_friction is μ × N = 0.3 × 9.8 N = 2.94 N.

4. Since F_gravity = F_friction, we have m × g = μ × N. Substituting the values, we get m × 9.8 m/s² = 0.3 × 9.8 N. The mass cancels out, leaving 9.8 m/s² = 0.3 × 9.8 m/s².

5. To find the angle θ, we can use the equation tanθ = h / 2.4384 m. Rearranging, we get h = 2.4384 m × tanθ.

Unfortunately, without the mass of the block, we cannot determine the exact height required. Please provide the mass of the block so that we can calculate the height accurately.