Accelerometer – Inertia Example Problem


An accelerometer is a device to measure acceleration. One of the simplest accelerometers is a small mass hanging from a thin rod or string that can pivot freely as a body accelerates. As the body containing the accelerometer accelerates one direction, the freely hanging weight will swing in the opposite direction. How far it swings is an indication of how much the body accelerates. This example problem shows how to use an accelerometer to determine the force of gravity.

Example Problem:
A rocket ship flies over the surface of a planet. There hangs a mass suspended by a wire to act as a simple accelerometer attached to the underside of the rocket. As the rocket accelerates at 10 m/s2, the accelerometer mass is deflected by an angle of 34°. What is the acceleration due to gravity (g) of this planet?

Solution: Here is an illustration of the problem. AccelerometerThe ship is accelerating to the right at a constant acceleration of 10 m/s². The mass is pulled down by the force of gravity, but held up by the tension in the wire. As the ship accelerates, the mass wants to stay in place but the tension in the wire pulls it along with the ship.

Here are the forces acting on the mass.

accelerometer forcesFirst, let’s find the forces acting in the x-direction.

ΣFx = Tsinθ

The acceleration is acting in the positive x-direction, so

ΣFx = ma.

Set these two to equal each other.

Tsinθ = ma

Now for the vertical forces.

ΣFy = Tcosθ – mg

Since the mass is not accelerating in the vertical direction, the sum of the forces vertically is equal to zero.

Tcosθ – mg = 0
Tcosθ = mg

Now we have two equations and two unknowns.

Tsinθ = ma
Tcosθ = mg

Divide the two equations into each other.

accelerometer math step 1
tanθ = a/g

Since we want to know the gravity on the planet, solve for g.

accelerometer math step 2

Plug in a = 10 m/s2 and θ = 34°

accelerometer math step 3
g = 14.8 m/s2

Answer:
The acceleration due to gravity on this planet is 14.8 m/s2.

Bonus:
If you’d like to know how many times Earth’s gravity, divide your answer by Earth’s gravity.

(14.8 m/s2) ÷ (9.8 m/s2) = 1.5

The gravity on this planet is 1.5x the force of gravity on Earth.


About Todd Helmenstine

Todd Helmenstine is the physicist/mathematician who creates most of the images and PDF files found on sciencenotes.org. Nearly all of the graphics are created in Adobe Illustrator, Fireworks and Photoshop. Todd also writes many of the example problems and general news articles found on the site.