Bodies Moving on Inclined Planes  Acting Forces
Required forces to move bodies up inclined planes.
If we neglect friction between the body and the plane  the force required to move the body up an inclined plane can be calculated as
F_{p} = W h / l
= W sin α
= m a_{g} sin α (1)
where
F_{p} = pulling force (N, lb_{f})
W = m a_{g}
= gravity force  or weight of body (N, lb_{f})
h = elevation (m, ft)
l = length (m, ft)
α = elevation angle (degrees)
m = mass of body (kg, slugs)
a_{g} = acceleration of gravity (9.81 m/s^{2}, 32.174 ft/s^{2})
By adding friction  (1) can be modified to
F_{p} = W (sin α + μ cos α)
= m a_{g} (sin α + μ cos α) (2)
where
Example  Pulling Force on Inclined Plane
A body with mass 1000 kg is located on a 10 degrees inclined plane. The pulling force without friction can be calculated as
F_{p} = (1000 kg) (9.81 m/s^{2}) sin(10°)
= 1703 N
= 1.7 kN
Online Inclined Plane Force Calculator  SI Units
The calculator below can be used to calculate required pulling force to move a body up an inclined plane.
Online Inclined Plane Force Calculator  Imperial Units
Angle of Repose
A body resting on a plane inclined at at an angle α to the horizontal plane is in a state of equilibrium when the gravitational force tending to slide the body down the inclined plane is balanced by an equal and opposite frictional force acting up the inclined plane.
For equilibrium the "angle of response" α can be expressed as:
μ = F_{p} / F_{n} = (W sin α) / (W cos α) = tan α (3)
Example  Gradient Force acting on a Car, Work Done and Power Required
The weight of a Tesla Model X is 2400 kg. The force acting on the car with 5% inclination can be calculated from (1) as
F_{p_5%} = (2400 kg) (9.81 m/s^{2}) sin (5°)
= 2051 N
The force acting on the car with 10% inclination can be calculated as
F_{p_10%} = (2400 kg) (9.81 m/s^{2}) sin (10°)
= 4088 N
If the Tesla is moving along the inclined roads with the same speed  the work done by the forces after 1 km can be calculated as
W_{5%} = (2051 N) (1000 m)
= 2051 kJ
W_{10%} = (4088 N) (1000 m)
= 4088 kJ
If the speed of the Tesla is 80 km/h (22.2 m/s)  the time to drive the distance can be calculated as
t_{80} = (1000 m) / (22.2 m/s)
= 45 s
The power required to move the vehicle (without rolling and air resistance) can be calculated as
P_{5%} = (2051 kJ) / (45 s)
= 45.6 kW
P_{50%} = (4088 kJ) / (45 s)
= 90.8 kW
If the speed of the car is reduced to 60 km/h (16.6 m/s)  the time to drive the distance can be calculated as
t_{60} = (1000 m) / (16.6 m/s)
= 60.2 s
The power required to move the vehicle (without rolling and air resistance) can be calculated as
P_{5%} = (2051 kJ) / (60.2 s)
= 34.1 kW
P_{50%} = (4088 kJ) / (60.2 s)
= 67.9 kW
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