A body of weight 500N rests on a plane inclined at 20 degrees to the horizontal. The coefficient of friction is 0.4, determine a force F at an angle of 15 degrees to the plane required to
(a) pull the body upwards
(b) push the body downwards
(c) pull the body downwards
(d) push the body upwards
To determine the force required to pull or push the body in different directions on the inclined plane, we need to consider the forces acting on the body.
Let's break down the forces acting on the body:
1. Weight (W): The weight of the body acts vertically downwards and is given as 500N.
2. Normal Force (N): The normal force is the force exerted by the inclined plane perpendicular to its surface. It acts perpendicular to the incline.
3. Friction Force (Ff): The friction force opposes the motion of the body and acts parallel to the inclined plane.
(a) To pull the body upwards:
In this case, we need to overcome the force of friction and the component of the weight acting parallel to the inclined plane. The force required is given by:
F = Ff + Fa
1. Considering the force of friction:
Ff = μ * N
where μ is the coefficient of friction, given as 0.4.
The normal force N can be determined using trigonometry:
N = W * cos(θ)
θ is the angle of inclination, given as 20 degrees.
2. Considering the component of the weight acting parallel to the inclined plane:
Fa = W * sin(θ)
Now, plug in the values and calculate:
N = 500N * cos(20°) ≈ 466.37N
Ff = 0.4 * 466.37N ≈ 186.55N
Fa = 500N * sin(20°) ≈ 170.96N
F = Ff + Fa
F = 186.55N + 170.96N ≈ 357.51N
Therefore, the force required to pull the body upwards is approximately 357.51N at an angle of 15 degrees to the plane.
(b) To push the body downwards:
In this case, we need to overcome the force of friction and the component of the weight acting parallel to the inclined plane. The force required is similar to case (a), and we can use the same values:
F = Ff + Fa ≈ 357.51N
Therefore, the force required to push the body downwards is approximately 357.51N at an angle of 15 degrees to the plane.
(c) To pull the body downwards:
In this case, we only need to consider the force of friction:
F = Ff ≈ 186.55N
Therefore, the force required to pull the body downwards is approximately 186.55N at an angle of 15 degrees to the plane.
(d) To push the body upwards:
In this case, we only need to consider the component of the weight acting parallel to the inclined plane:
F = Fa ≈ 170.96N
Therefore, the force required to push the body upwards is approximately 170.96N at an angle of 15 degrees to the plane.