A body of weight 500N is lying on a rough plane inclined at angle 3.2 with the horizontal. It is supported by an effort p, parallel to the plane. Determine the minimum and maximum value of P, for which the equilibrium can exist if the angle of friction is 20
To determine the minimum and maximum values of the effort P, we need to consider the forces acting on the body and apply the conditions for equilibrium.
The weight of the body, 500N, acts vertically downward. We can resolve this force into two components: one parallel to the inclined plane and the other perpendicular to it.
The component parallel to the inclined plane is given by: W_parallel = Weight * sin(angle of inclination) = 500N * sin(3.2°).
The component perpendicular to the inclined plane is given by: W_perpendicular = Weight * cos(angle of inclination) = 500N * cos(3.2°).
Now, let's consider the forces acting on the body along the inclined plane:
1. The supporting force or effort P acting parallel to the plane.
2. The frictional force acting against the motion of the body.
The frictional force can be determined using the formula: Frictional Force = Coefficient of Friction * Normal Force.
The normal force acting perpendicular to the inclined plane is given by: Normal Force = W_perpendicular = 500N * cos(3.2°).
The coefficient of friction can be determined using the given angle of friction, which is 20°.
Now, we can calculate the minimum and maximum values of P:
1. Minimum Value of P:
In order to maintain equilibrium, the applied effort P must be at least equal to the frictional force. So, the minimum value of P is equal to the frictional force, which is given by: Frictional Force = Coefficient of Friction * Normal Force.
2. Maximum Value of P:
In this case, we consider the maximum value of P where the body is on the verge of sliding down the inclined plane. The frictional force is now opposing the motion of the body, so the maximum value of P is equal to the frictional force multiplied by the other components of the force triangle. The frictional force is given by: Frictional Force = Coefficient of Friction * Normal Force.
To summarize:
- Minimum Value of P = Frictional Force = Coefficient of Friction * Normal Force.
- Maximum Value of P = Frictional Force * other components of the force triangle.