Question

suppose you are designing a cart for carrying goods downhill. to maximize cart speed should you design wheels so that their moment of inertia about their rotation axes are large or small or it does not matter?explain in detail assuming that mechanical energy is conserved.

Answer 1

Answer on Question #39928 – Physics - Mechanics | Kinamatics | Dynamics

Suppose you are designing a cart for carrying goods downhill. To maximize cart speed should you design wheels so that their moment of inertia about their rotation axes are large or small or it does not matter? Explain in detail assuming that mechanical energy is conserved.

The conservation law of mechanical energy:

$$T + U = \text{const}$$

or:

$$T_1 + U_1 = T_2 + U_2$$

For state (1):

$$T_1 = 0, U_1 = \text{mgh}$$

For state (2):

$$U_2 = 0, T_2 = \frac{mv^2}{2} + \frac{lw^2}{2}$$ $$v - \text{cart' speed}$$

$l$ – sum of moments of inertia all wheels

$\omega$ – angular velocity of wheels

$$\omega = \frac{v}{r}, \text{where } r - \text{radius of wheel}$$

Therefore:

$$\text{mgh} = \frac{mv^2}{2} + \frac{l \left(\frac{v}{r}\right)^2}{2} = \frac{mv^2}{2} \left(1 + \frac{l}{mr^2}\right)$$ $$v = \sqrt {\frac {2 g h}{\left(1 + \frac {I}{m r ^ {2}}\right)}}$$

To maximize the cart speed, you should design the wheels so that their moments of inertia about their rotation axes are small as it is possible. In ideal case $I \ll mr^2$ or $\frac{I}{mr^2} \ll 1$, so we can ignore $\frac{I}{mr^2}$, because it's small compared with 1. That's mean:

$$v _ {m a x} = \sqrt {2 g h}$$

Answer:

$$v _ {m a x} = \sqrt {2 g h}$$