# Inclined Plane Calculator

Created by Davide Borchia
Last updated: Aug 16, 2022

You probably didn't even think that ramps are machines: our inclined plane calculator will guide you on a quick but comprehensive tour of formulas and equations for an inclined plane, even considering friction.

What are you going to find in our inclined plane calculator?

• What is an inclined plane?
• The forces on an inclined plane: the force of friction and gravitational force components;
• How to find the acceleration down a ramp;
• The physics of an inclined plane: how to find the time of descent and the final speed;
• The behavior of bodies rotating down an inclined plane.

## What is an inclined plane: the physics of ramps

An inclined plane is exactly what the name suggests: a flat surface that lies at a certain angle with respect to the horizontal plane. Inclined planes, or ramps, appear everywhere: since they give a mechanical advantage, they are used when you need to raise or lower a heavy object (too heavy to lift vertically) or to help reduce the effort while climbing stairs or small obstacles.

🙋 Ramps are not only widespread in space but in time too. Most archeologists agree on the fact that the only way Ancient Egyptians could build their colossal pyramids.

An inclined plane is characterized by its height and length, which together define the angle of the ramp. If you want the math, here is the equation to calculate the angle of inclination of a ramp:

$\sin{(\theta)} = \frac{H}{L}$

Where:

• $H$ is the height;
• $L$ is the length; and
• $\theta$ is the angle of the inclined plane.

## The equations for the inclined plane

On an inclined plane, the weight of an object is decomposed in two components, whose proportions depend on the angle of inclination: we calculate them with the following characteristic equations for an inclined plane:

\begin{align*} F_{\parallel}&=m\cdot g\cdot\sin{(\theta)}\\ F_{\perp}&=m\cdot g\cdot\cos{(\theta)} \end{align*}

Where:

• $F_{\parallel}$ is the **component of the gravitational force parallel to the plane's surface;
• $F_{\perp}$ is the component of the gravitational force perpendicular to the surface;
• $m$ is the mass of the body; and
• $g$ the acceleration due to gravity.

These formulas come from a bit of trigonometry. If you calculate the vector sum of the two components, you would find the value of the gravitational force acting on the body.

If you add friction to the problem, you slightly complicate things. If the body moves down (descending the ramp), the friction force points upward (against the motion direction). Remember the formula for friction:

$F_{\text{f}} = f\cdot m\cdot g\cos{(\theta)}$

To calculate an inclined plane's friction force, we must consider the perpendicular component of the gravitational force.

When computing the total force acting on an object sliding down an inclined plane, we calculate the sum of the forces parallel to the surface. Perpendicular forces don't interfere directly with the motion (but are implied in the calculations for the force of friction on the inclined plane). The total force is:

\begin{align*} F& = F_{\parallel}-F_{\text{f}}\\ &= m\cdot g \left(\sin{(\theta)} - f\cdot\cos{(\theta)}\right) \end{align*}

🙋 If the ramp is not inclined enough, the friction force may be high enough to surpass the parallel component of the gravitational force: in this case, the body would not move!

## Motion on an inclined plane: formula for acceleration on an inclined plane, time of descent and final speed

Once you know the resulting force on the body, you can apply Newton's second law to compute the acceleration on an inclined plane. In the absence of other forces acting on the body, we calculate the acceleration as:

$a = \frac{F}{m}$

Using the equation for a uniformly accelerated linear motion, we find both the time of descent and the final speed.

\begin{align*} t &= \frac{\left(\sqrt{v_0^2 -+ 2\cdot L\cdot a} - v_0\right)}{a}\\ v_t & = v_0+a\cdot t \end{align*}

Where:

• $t$ is the time;
• $v_0$ is the initial speed, and $v_t$ the final speed at the time $t$; and
• $a$ the acceleration on the inclined plane.

## Rolling down an inclined planed: the physics of ramps and round objects

The equations we saw in the previous paragraph are valid for sliding objects, like a cube. Our inclined plane calculator allows you to calculate acceleration, time, final speed (and more) for objects rolling down a ramp.

The analysis of the motion would be too complex to do in the same fashion: we will resort to the conservation of energy to find the acceleration on the inclined plane. The formula, which takes into account torque and rotational energy, is:

$a = F_{\parallel} \Big/ \left(m + \frac{I}{r^2}\right)$

Where:

• $I$ is the mass moment of inertia; and
• $r$ the radius (the distance between the center of the rolling body and the surface of the plane).

You will find many choices of solids in the first variable of our inclined plane calculator.

Davide Borchia
Select object and initial parameters
Object
Cubic block
Mass (m)
kg
Angle (θ)
deg
Friction coefficient (f)
Height (H)
m
Results
Acceleration (a)
m/s²
Sliding time (t)
sec
Final velocity (v)
m/s
Energy loss (ΔE)
J
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