A body of weight 500N is liying on a rough plane inclined at angle 25 with the horizontal, it ir supported.by an effort P, parallel to the plane.determine the minimum and maximum value of P, for the equilibriun can exist if the angle of friction is 20
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To determine the minimum and maximum values of the effort P for equilibrium to exist, we need to consider the forces acting on the body on the inclined plane.
1. Resolve the weight into its components:
The weight (W) can be resolved into two components: one perpendicular to the plane (W⊥) and one parallel to the plane (W∥).
W⊥ = W * cos(θ) = 500 N * cos(25°) ≈ 450.86 N
W∥ = W * sin(θ) = 500 N * sin(25°) ≈ 213.94 N
2. Determine the minimum value of P:
The minimum value of P occurs when the body is just on the verge of sliding down the inclined plane. In this case, the force of friction (F) is at its maximum value.
F = μ * W⊥ = 0.2 * 450.86 ≈ 90.17 N
The minimum value of P is equal to the force of friction, so Pmin = F ≈ 90.17 N
3. Determine the maximum value of P:
The maximum value of P occurs when the body is just on the verge of sliding up the inclined plane. In this case, the force of friction (F) is at its minimum value, which is zero.
F = 0
To find the maximum value of P, we need to consider the forces perpendicular to the plane:
ΣF⊥ = W⊥ - P * cos(θ) = 0
P * cos(θ) = W⊥
Pmax = W⊥ / cos(θ) = 450.86 / cos(25°) ≈ 494.82 N
Therefore, the minimum value of P is approximately 90.17 N, and the maximum value of P is approximately 494.82 N.
To determine the minimum and maximum value of the effort P for equilibrium to exist, we need to consider the forces acting on the body on the inclined plane. In this case, the forces involved are the weight of the body (500N) and the frictional force.
1. Determine the components of the weight force:
The weight force acting on the inclined plane can be resolved into two components: perpendicular to the plane (N) and parallel to the plane (Wx).
The perpendicular component (N) can be calculated using the equation N = mgcos(θ), where m is the mass of the body and θ is the angle of inclination. In this case, θ = 25°, so N = 500N * cos(25°).
The parallel component (Wx) can be calculated using the equation Wx = mgsin(θ), where m is the mass of the body and θ is the angle of inclination. In this case, θ = 25°, so Wx = 500N * sin(25°).
2. Determine the maximum frictional force:
The maximum frictional force (Fmax) can be calculated using the equation Fmax = μN, where μ is the coefficient of friction and N is the perpendicular component of the weight force.
In this case, μ = tan(20°) because the angle of friction is given as 20°. So, Fmax = tan(20°) * N.
3. Determine the minimum value of the effort P:
The minimum value of the effort P occurs when the frictional force is at its maximum value. In this case, the minimum value of P will be equal to the maximum frictional force, which is Fmax = tan(20°) * N.
4. Determine the maximum value of the effort P:
The maximum value of the effort P occurs when the frictional force is at its minimum value, which is zero. This happens when the body is just at the verge of slipping. So, the maximum value of P in this case is equal to the weight force parallel to the plane, which is Wx.
Therefore, the minimum value of the effort P is tan(20°) * N, and the maximum value of the effort P is Wx.