# Moment of Inertia for a car

I'm in a physics class in which we calculated the moment of inertia of a ball rolling down a ramp using a conservation of energy equation: $${1 \over2}mv^2 + {1 \over2}Iω^2 = mgh$$

where $$I$$ is the moment of inertia of the ball.

In a different class I'm in, I am designing a toy car. For some of our simulations, we need to determine the moment of inertia for the car. Could it be possible to determine $$I$$ for the car using a conservation of energy equation like the one above?

• Hopefully, you will never have to deal with moment of inertia in a real car! Dec 10, 2019 at 18:23
• Yes, but It would be the moment of inertia of the rotating parts of the car, and not of the entire car. Dec 10, 2019 at 22:45

I guess you need the moment of inertia of the car itself and not the wheels. Therefore you could determine this using the formula for a compound pendulum $$T=2\pi \sqrt{\frac{I}{mgR}}$$ and doing the experiment. For the experiment you need to hang your car from a rope or something similar and let it swing, measure the period $$T$$ and the radius $$R$$ to the center of the car and measure the mass $$m$$ of your car. If you put these values into the formula and solve for $$I$$ you get the needed moment of inertia.
A rough cut estimate for a passenger vehicle's moment of inertia about the axis normal to the pavement is $$m \cdot a \cdot b$$, where $$m$$ is mass, $$a$$ is the distance from center of mass to front axle, and $$b$$ is the distance from center of mass to rear axle.
• Yes, this is the lumped mass model, where each axle bears a certain percentage of the total mass and the MMOI is $$I = m_a a^2 + m_b b^2$$ by using \begin{aligned} m_a & = \frac{b}{a+b} m \\ m_b & = \frac{a}{a+b} m \end{aligned} you arrive at your formula. It would be nice if you included this calculations in your answer in order to enlighten the reader. May 10, 2022 at 19:25