A 20 kg slab is to be pulled up a plane inclined 20 degrees with the horizontal at a constant speed by a force that is directed 15 degrees above the surface of the inclined plane. Determine the magnitude of the force necessary to perform the task. The coefficient of friction between the block and the plane is 0.2.
Ws = M*g = 20 * 9.8 = 196 N.
Fp = 196*sin20 = 67 N.
Fn = 196*Cos20 - F*sin15 = 184-0.26F.
Fs = 0.2(184-0.26F) = 36.8-0.052F. = Static friction.
F*Cos15-Fp-Fs = M*a.
0.97F-67-(36.8-0.052F) = M*0 = 0.
0.97F-67-36.8+0.052F = 0,
1.022F = 36.8, F = 36 N.
In the 0.97F-67-(36.8-0.052F) = M*0 = 0, where the 0.97 came from?
Cos15 = 0.966.
To determine the magnitude of the force necessary to pull the slab up the inclined plane, we need to consider the forces acting on the slab and then apply Newton's second law of motion.
1. Identify the forces acting on the slab:
- The weight of the slab, acting vertically downwards, can be calculated using the formula: weight = mass × gravitational acceleration.
- The normal force, acting perpendicular to the inclined plane.
- The force of friction, acting parallel to the inclined plane and opposing the motion of the slab.
2. Decompose the force that is directed 15 degrees above the surface of the inclined plane into horizontal and vertical components:
- The horizontal component of the force will help overcome the force of friction.
- The vertical component of the force will counteract a fraction of the weight.
3. Calculate the weight of the slab:
- Weight = mass × gravitational acceleration = 20 kg × 9.8 m/s².
4. Determine the normal force:
- The normal force is equal to the component of the weight perpendicular to the inclined plane, which can be calculated as follows:
Normal force = weight × cos(θ), where θ is the angle of inclination (20 degrees).
5. Determine the force of friction:
- The force of friction can be calculated using the formula: force of friction = coefficient of friction × normal force.
6. Calculate the opposing force due to friction:
- Opposing force = force of friction × cos(θ).
7. Determine the horizontal component of the force:
- Horizontal component = force × cos(15 degrees).
8. Calculate the net force in the horizontal direction:
- Net horizontal force = horizontal component of the force - force of friction.
9. Apply Newton's second law of motion:
- Net horizontal force = mass × acceleration.
- Since the slab is moving at a constant speed, the acceleration is zero.
10. Lastly, solve for the magnitude of the force:
- Magnitude of the force = net horizontal force.
By following these steps, you can calculate the magnitude of the force necessary to pull the slab up the inclined plane.