# Does the total mass of a wheeled object affect acceleration down an incline?

For an object such as a roller coaster which sits on wheels which undergo rolling without slipping, does changing the total mass of the vehicle change how it accelerates down the incline? I have been studying these kind of scenarios lately and my calculations tell me that a larger mass will result in a larger acceleration but many online sources claim that this is false. I intend to test this myself when I can but until then this is the best I can do to find an answer.

Angular acceleration "α" = acceleration / radius (for the wheels).

Force of static friction on each wheel = kma where "k" is the coefficient for the moment of inertia. I got this equation from considering static friction acting at a distance "r" causing a torque which is equal to Fr = km(r^2)α and simplifying.

So nkma - MgSin(theta) = -Ma where "n" is the number of wheels and "M" is the total mass with "m" being the mass of a wheel. I took static friction as positive pointing up and made "Ma" negative to indicate the downward acceleration.

Solving for the acceleration I find that you get:

a = MgSin(theta)/(nkm + M)

I ran some calculations to see how this plays out and found that although changing the total mass will increase the acceleration, the change is quite small unless the change in total mass is quite large. I'm assuming everything else is equal of course such as number of wheels, the value of "k" and angle of incline. I'm also ignoring air resistance and other resistances such as rolling resistance.

So am I right here or have I made another mistake?

## 1 Answer

Let's ignore the energy stored in the rotation of the wheels for a second, in this case mgh=0.5mv^2 due to conservation of energy and v then does not depend on mass. (If you drop a 1kg mass and a 10kg mass of the same size off the Eiffel tower they both hit the ground at the same time.)

You could devise a roller coaster with huge wheels with lots of mass, it would go down much slower and when it got to the bottom energy (mgh) is still conserved but now it is the sum of 0.5 mv^2 plus the energy store in the wheels rotating. You could invent a really slow roller coaster, and it would still make it to the top of the next hill.

• The point of my question is to compare the acceleration of two roller coasters (or any similar object such as a car) where the only difference is their total mass and everything else is equal, inlcuding the size and mass of the wheels. – An_African_Ape Oct 23 '18 at 14:46
• If the wheels are the same then the coaster with the heavier load will go slightly slightly faster as the wheel's mass are a smaller proportion compared to total mass. – PhysicsDave Oct 23 '18 at 15:08