A playground ride consists of a disk of mass M = 41 kg and radius R = 1.7 m moun

Question

A playground ride consists of a disk of mass M = 41 kg and radius R = 1.7 m mounted on a low-friction axle. A child of mass m = 16 kg runs at speed v = 2.1 m/s on a line tangential to the disk and jumps onto the outer edge of the disk.

ANGULAR MOMENTUM
(a) Consider the system consisting of the child and the disk, but not including the axle. Which of the following statements are true, from just before to just after the collision?

The angular momentum of the system about the axle hardly changes.

The torque exerted by the axle is zero because the force exerted by the axle is very small.

The axle exerts a force on the system but nearly zero torque.

The momentum of the system doesn't change.

The angular momentum of the system about the axle changes.

The momentum of the system changes

Increased translational kinetic energy of the disk.Increased thermal energy of the disk and child.    Increased chemical energy in the child.

Explanation / Answer

part a:

torque=perpendicular distance*force

as there is no friction involved, angular momentum is conserved.

hence below statements are correct:

the torque exerted by the axle is nearly zero even though the force is large, because |r| is nearly zero.

the angular momentum of the system about the axle hardly changes.

the axle exerts a force on the system but nearly zero torque.

part b:

magnitude of angular momentum of the child beforce collision=moment of inertia*angular speed

=mass*distance^2*(linear speed/distance)

=mass*linear speed*distance

=16*2.1*1.7

=57.12 kg.m^2/s

part c:

as angular momentum is conserved, angular momentum of the child plus disk system=angular momentum of the system before collision

=57.12 kg.m^2/s

part d:

let angular speed of the disk is w.

angular momentum of the system of child plus disk just after the collision

=moment of inertia of the child*angular speed of the child + moment of inertia of the disk*angular moment of the disk

=mass*distance^2*angular speed + 0.5*mass*radius^2*angular speed

=16*1.7^2*w+0.5*41*1.7^2*w

=105.485*w

using conservation of angular momentum,

105.485*w=57.12

==>w=0.5414 rad/s

=5.17 rpm

part e:

time taken to complete one revolution

=1 revolution/angular speed

=1 revolution /(5.17 revolution/minute)

=11.6 seconds

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