A block weighing 71.5 N rests on a plane inclined at 24.1° to the horizontal. The coefficient of the static and kinetic frictions are 0.26 and 0.13 respectively. What is the minimum magnitude of the force F, parallel to the plane, that will prevent the block from slipping?
See previous post: Thu,2-20-14,5:05 PM.
To find the minimum magnitude of the force F that will prevent the block from slipping, we need to consider the forces acting on the block.
1. Normal Force (N): The component of the weight acting perpendicular to the plane is the normal force. It can be determined using the formula N = mg * cos(θ), where m is the mass of the block, g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of inclination in radians.
2. Friction Force (Ff): The friction force opposes the motion of the block and can be calculated using the formula Ff = μ * N, where μ is the coefficient of friction.
3. Force F: This is the force parallel to the plane that prevents the block from slipping. It acts in the same direction as the applied force but opposite to the friction force.
To find the minimum magnitude of force F, we need to balance the forces on the block along the inclined plane:
F - Ff = m * a,
where a is the acceleration of the block along the inclined plane. Since the block is not slipping, the acceleration a is zero.
Therefore, we have:
F - Ff = 0,
F = Ff.
To find the friction force Ff, we use the coefficient of static friction as the block is at rest:
Ff = μ(static) * N.
First, calculate the normal force:
N = mg * cos(θ).
Then, calculate the friction force:
Ff = μ(static) * N.
Finally, the minimum magnitude of the force F that will prevent the block from slipping is equal to the friction force Ff.
Let's calculate it.
Given:
Weight of the block (W) = 71.5 N
Angle of inclination (θ) = 24.1°
Coefficient of static friction (μ(static)) = 0.26
1. Calculate the normal force (N):
N = W * cos(θ).
2. Calculate the friction force (Ff):
Ff = μ(static) * N.
3. The minimum magnitude of the force F:
F = Ff.
Calculating step by step:
1. Normal force (N):
N = 71.5 N * cos(24.1°).
Using the calculator:
N ≈ 64.08 N.
2. Friction force (Ff):
Ff = 0.26 * 64.08 N.
Using the calculator:
Ff ≈ 16.66 N.
3. Minimum magnitude of the force F:
F ≈ 16.66 N.
Therefore, the minimum magnitude of the force F, parallel to the plane, that will prevent the block from slipping is approximately 16.66 N.