# Inclined Plane Calculator

Inclined plane calculator in physics to find friction force, normal force, tension force and acceleration of an object on an inclined plane which is attached to a pulley with a hanging mass. It's assumed that the string has negligible mass and pulley is frictionless. There is a friction between the object and the plane. Maximum frictional force when the object is not moving and in static condition:

$${ F }_{ fmax }={ \mu }_{ s }{ m }_{ 1 }g\cos { \alpha }$$

Formulas for object-1 just about to start motion in +x direction:

Newton’s second law for object-2:

$$T={ m }_{ 2 }g$$

Newton’s second law for object-1:

$$T-{ m }_{ 1 }g\sin { \alpha } -{ F }_{ fmax }=0$$ $${ m }_{ 2 }g-{ m }_{ 1 }g\sin { \alpha } -{ F }_{ fmax }=0$$ $${ m }_{ 2 }g={ m }_{ 1 }g\sin { \alpha } +{ F }_{ fmax }$$

If following condition is met, than the object-1 starts to move in +x direction.

$${ m }_{ 2 }g>{ m }_{ 1 }g\sin { \alpha } +{ F }_{ fmax }$$

Formulas for object-1 just about to start motion in -x direction:

Newton’s second law for object-2:

$$T={ m }_{ 2 }g$$

Newton’s second law for object-1:

$$T+{ F }_{ fmax }-{ m }_{ 1 }g\sin { \alpha } =0$$ $${ m }_{ 2 }g+{ F }_{ fmax }-{ m }_{ 1 }g\sin { \alpha } =0$$ $${ m }_{ 2 }g={ m }_{ 1 }g\sin { \alpha } -{ F }_{ fmax }$$

If following condition is met, than the object-1 starts to move in -x direction.

$${ m }_{ 2 }g<{ m }_{ 1 }g\sin { \alpha } -{ F }_{ fmax }$$

Formulas for object-1 that is accelerating in +x direction

Maximum frictional force when the object is moving:

$${ F }_{ fmax }={ \mu }_{ k }{ m }_{ 1 }g\cos { \alpha }$$

Newton’s second law for object-1:

$$T{ -m }_{ 1 }g\sin { \alpha } -{ \mu }_{ k }{ m }_{ 1 }g\cos { \alpha } ={ m }_{ 1 }a$$

Newton’s second law for object-2:

$${ m }_{ 2 }g-T={ m }_{ 2 }a$$

If these two equations are solved together, than

$$a=\frac { { m }_{ 2 }g-{ m }_{ 1 }gsin\alpha -{ \mu }_{ k }{ m }_{ 1 }g\cos { \alpha } }{ { m }_{ 1 }+{ m }_{ 2 } }$$

Formulas for object-1 that is accelerating in -x direction:

Newton’s second law for object-1:

$${ m }_{ 1 }g\sin { \alpha }-T -{ \mu }_{ k }{ m }_{ 1 }g\cos { \alpha } ={ m }_{ 1 }a$$

Newton’s second law for object-2:

$$T{ -m }_{ 2 }g={ m }_{ 2 }a$$

If these two equations are solved together, than

$$a=\frac { { -m }_{ 2 }g+{ m }_{ 1 }gsin\alpha -{ \mu }_{ k }{ m }_{ 1 }g\cos { \alpha } }{ { m }_{ 1 }+{ m }_{ 2 } }$$

### Object on Inclined Plane with Static and Kinetic Friction Calculator : INPUT PARAMETERS Mass-1 [m1] g kg oz lb Mass-2 [m2] Gravity [g] m/s^2 g m/h^2 cm/s^2 in/s^2 km/h^2 miles/h^2 ft/s^2 Angle [α] deg rad Static friction coefficient [µs] --- Kinetic friction coefficient [µk]

Note: Use dot "." as decimal separator.

 RESULTS Status Tension [T] N kN lbf Normal Force [N] Friction Force [Ffmax] Friction Type --- Acceleration [a] m/s^2 g m/h^2 cm/s^2 in/s^2 km/h^2 miles/h^2 ft/s^2 ADDITIONAL RESULTS Angle [α] at which the object starts to slide in +x direction for given m1, m2, µs deg rad Angle [α] at which the object starts to slide in -x direction for given m1, m2, µs

Note: Default rounding is 7 decimal places.

### Box Sliding Incline Example: The box shown has mass 5 kg and lies on a plane tilted at an angle of 34° to the horizontal. What is the speed of the object at the bottom of the inclined plane if the box starts from the rest 1.25m up the plane from its base?

Solution:

 INPUT PARAMETERS Mass-1 [m1] 5 kg Mass-2 [m2] 0 Gravity [g] 9.8 m/s^2 Angle [α] 34 deg Static friction coefficient [µs] 0.40 --- Kinetic friction coefficient [µk] 0.35
 RESULTS Status Object accelerates in -x direction Tension 0 N Acceleration 2.6364916 m/s^2

Go to Acceleration Calculator page to calculate the speed at the bottom of the inclined plane.

 KNOWN PARAMETERS Acceleration, Initial Velocity, Displacement INPUT PARAMETERS Displacement (∆x) 1.25 m Initial Velocity (V0) 0 m/s Final Velocity (Vf) - Elapsed Time (t) - s Constant Acceleration (a) 2.63649 m/s^2

 RESULTS Parameter Solution Unit Displacement 1.25 m Distance 1.25 Initial Velocity 0 m/s Final Velocity 2.56734 Average Velocity 1.28367 Average Speed 1.28367 Constant Acceleration 2.63649 m/s^2 Time 0.97377 s

The speed at the bottom of inclined plane is 2.56734 m/s.