Effort Required to Move the Body Up an Inclined Plane
The effort (P) required to move the body up an inclined plane as shown in Fig. 1.20, is given by
![](data:image/png;base64,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)
where
W = Weight of the body,
α = Angle of inclination of the plane with the horizontal,
θ = Angle which the line of action of P makes with the weight of the body W, and
Ï• = Angle of friction.
The effort (P) required to move the body up an inclined plane as shown in Fig. 1.20, is given by
where
W = Weight of the body,
α = Angle of inclination of the plane with the horizontal,
θ = Angle which the line of action of P makes with the weight of the body W, and
Ï• = Angle of friction.
Notes :
1. When friction is neglected, then Ï• = 0, In that case,
2. When effort P is applied horizontally, then θ = 90°. In that case
3. When effort P is applied parallel to the plane, then θ = 90° + α. In that case,
Effort Required to Move the Body Down an Inclined Plane
The effort (P) required to Move the body down an inclined plane as shown in Fig. 1.21, is given by
Notes:
1. When friction is neglected, then Ï• = 0. In that case,
2. When effort is applied horizontally, then θ = 90°. In That case,
3. When effort is applied parallel to the plane, then θ = 90° + α. In that case,
Efficiency of an Inclined Plane
It is defined as the ratio of the effort required neglecting friction ( i.e. Po) to the effort required considering friction (i.e. P). Mathematically, efficiency of an inclined plane,
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