Question

(someone tried to answer thisquestion but it was wrong, can someone please try to answer thisquestion!!!!) thankyou!!!Gearsare important components in many mechanical devices, frommechanical clocks to bicycles. In fact, they are present whenever amotorproduces rotational motion.Anexample of a simple gear system is shown in thefigure.The bigger wheel (wheel 1) has radius $r_1$ , while the smaller one (wheel 2) has radius $r_2$ . The two wheels have small teeth and are connectedthrough a metal chain so that when wheel 1rotates, the chain moveswith it and causes wheel 2 to rotate as well.
1.)Let wheel 1rotate at a constant angular speed $omega_1$ . Find the ratio $frac{omega_1}{omega_2}$ ofthe angular speed of wheel 1 to the angularspeed of wheel 2.Expressyour answer in terms of any or all of the variables $r_1$ and $r_2$ . $frac{omega_1}{omega_2}$ = ?2.) The rotation of wheel 1 is caused by atorque $tau_1$ . Find the ratio $frac{tau_1}{tau_2}$ ofthe torque acting on wheel 1 to the torqueacting on wheel 2.Express your answer in terms of anyor all of the variables $r_1$ and $r_2$ . $frac{tau_1}{tau_2}$ = ?3.)If the power needed to rotate wheel 1 is $P_1$ , what is the ratio of the power of wheel 1 to the power of wheel2?Express your answer in terms of anyor all of the variables $r_1$ and $r_2$ .= ?

Answer 1

since the chain connects the two gears then every point on thechain is moving at the same velocity so the tangential velocity soall of the teeth of bothgears have to be moving at the same speed.
Since Vtan=rω and Vtan of the first gear is equalto the Vtan of the second you solve forω1/ω2 whichwould be r2/r1
SinceF1=F2 andτ=Fr , you solveforτ1/τ2 which would be r1/r2
Then the P=τω comes out to be (r1/r2)(r2/r1)=1