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 0.342
Answer kitna hsi
To determine the minimum and maximum value of the effort P, we first need to analyze the forces acting on the body.
1. Resolve the weight force into two components:
- The component parallel to the inclined plane: W_parallel = W * cos(θ)
- The component perpendicular to the inclined plane: W_perpendicular = W * sin(θ)
2. Determine the friction force (F_friction) acting on the body. Since the body is at maximum equilibrium, the force of friction will act in the opposite direction to the effort P.
F_friction = μ * W_perpendicular
where μ is the coefficient of friction, given as 0.342, and W_perpendicular is the component of weight perpendicular to the inclined plane.
3. Calculate the minimum value of the effort P:
For minimum equilibrium, the friction force needs to be equal to the maximum possible value to prevent the body from sliding.
Therefore, P_min = F_friction
4. Calculate the maximum value of the effort P:
For maximum equilibrium, the friction force needs to be equal to the minimum possible value to allow the body to slide.
Therefore, P_max = F_friction + μ * W_parallel
Now, let's substitute the given values and calculate the minimum and maximum values of the effort P.
Given:
Weight, W = 500N
Angle of inclination, θ = 3.2°
Coefficient of friction, μ = 0.342
1. Determine the components of weight:
W_parallel = W * cos(θ)
= 500 * cos(3.2°)
W_perpendicular = W * sin(θ)
= 500 * sin(3.2°)
2. Calculate the friction force:
F_friction = μ * W_perpendicular
3. Calculate the minimum value of the effort P:
P_min = F_friction
4. Calculate the maximum value of the effort P:
P_max = F_friction + μ * W_parallel
Now, you can substitute the calculated values to find the minimum and maximum values of the effort P.
To determine the minimum and maximum value of P for which equilibrium can exist, we need to consider the forces acting on the body.
1. Draw a free body diagram:
- The weight (W) acts vertically downwards with a magnitude of 500N.
- The normal force (N) acts perpendicular to the inclined plane.
- The friction force (F) opposes the motion and acts parallel to the inclined plane.
- The applied force or effort (P) acts parallel to the inclined plane in the same direction as the friction force.
2. Resolve the weight and normal force:
The weight can be resolved into two components:
- The component parallel to the inclined plane is Wsinθ.
- The component perpendicular to the inclined plane is Wcosθ.
The normal force is equal to the perpendicular component of the weight, so N = Wcosθ.
3. Determine the maximum friction force:
The maximum friction force (F_max) can be calculated using the equation F_max = μN, where μ is the coefficient of friction.
Given that the angle of friction is 0.342, we can use the relation μ = tan(θ_f), where θ_f is the angle of friction.
So, μ = tan(0.342).
4. Analyze the forces in equilibrium:
For the body to be in equilibrium, the sum of the forces parallel to the inclined plane must be zero.
Sum of forces parallel to the inclined plane:
P - F = 0
Since F_max = μN, we have:
P - μN = 0
Substituting N = Wcosθ, we get:
P - μWcosθ = 0
5. Calculate the minimum and maximum values of P:
To determine the minimum and maximum values of P, we need to consider the limiting angles of friction.
The minimum value of P occurs when the angle of friction is maximum (θ_f).
So, P_min = μWcosθ_f
The maximum value of P occurs when the angle of friction is minimum (θ_f).
So, P_max = μWcosθ_f
Substituting the given values into the equations, you can calculate the minimum and maximum values of P.