Question

Most gasoline engines in today's automobiles are belt driven. This means that the crankshaft, a rod which rotates and drives the pistons, is timed to the camshaft, the mechanism which actuates the valves, by means of a belt. Starting from rest, assume it takes t = 0.0980 s for a crankshaft with a radius of r1 = 3.75 cm to reach 1350 rpm. If the belt doesn't stretch or slip, calculate the angular acceleration of the larger camshaft, which has a radius of r2 = 7.50 cm, during this time period.

Answer 1

Angular velcoity of larger camshatt:

$$ a_{1}=1350\left(\frac{2 \pi}{60}\right) \mathrm{rad} / \mathrm{s}=141.37 \mathrm{rad} / \mathrm{s} $$

Angular accleration:

$$ \begin{aligned} \alpha_{1} &=\frac{a_{1}}{t} \\ &=\frac{141.37 \mathrm{rad} / \mathrm{s}}{0.0980 \mathrm{~s}} \\ &=1442.568 \mathrm{rad} / \mathrm{s}^{2} \end{aligned} $$

As there is no slipping, the torque on camshaft 1 is equal totorque on cam shaft 2

$$ \begin{aligned} \alpha_{1} r_{1} &=\alpha_{2} r_{2} \\ 1442.568 \mathrm{rad} / \mathrm{s}^{2}(3.75 \mathrm{~cm}) &=\alpha_{2}(7.50 \mathrm{~cm}) \end{aligned} $$

Solve for angular acceleration of cam shaft 2 :

$$ \alpha_{2}=721.284 \mathrm{rad} / \mathrm{s}^{2} $$

Round off the result to 3 significant digits

$$ \alpha_{2}=721 \mathrm{rad} / \mathrm{s}^{2} $$