Question

Before the invention of a wheel turning on an axle, ancient people moved heavy loads by placing rollers under them. (Modern people use rollers, too. Any hardware store will sell you a roller bearing for a lazy susan.) A stone block of mass 672 kg moves forward at 0.319 m/s, supported by two uniform cylindrical tree trunks, each of mass 82.0 kg and radius 0.343 m. No slipping occurs between the block and the rollers or between the rollers and the ground. Find the total kinetic energy of the moving objects.

Answer 1

Answer:$K.E.=36.277+1.0429=37.319 J$

Explanation:

Velocity of Top most point of wheel is twice the Velocity of centre of mass of wheel

$i.e V_{wheel}=2V_{cm}=0.319 m/s$

Thus angular velocity is given by

$\omega =\frac{V_{cm}}{R}$

$\omega =0.465 rad/s$

$Kinetic\ Energy=Rotational + transtational$

$K.E.=\frac{1}{2}m_{stone}V^2+2\left [ \frac{1}{2}m_{cyl}\left [ \frac{V}{2}\right ]^2\right ]+2\left [ \frac{1}{2}I\omega ^2\right ]$

$K.E.=\frac{V^2}{2}\left [ m_{stone}+\frac{m_{roller}}{2}\right ]+I\omega ^2$

$K.E.=\frac{0.319^2}{2}\left [ 672+\frac{82}{2}\right ]+\frac{82\times 0.343^2\times 0.465^2}{2}$

$K.E.=36.277+1.0429=37.319 J$