Question

A man of mass M=50 kg standing on the edge of a platform (moment of inertia of the platform is I and radius R=1 m) is rotating in anticlockwise direction at an angular speed of 20 rad/s. The man starts walking along the rim with a speed 1 m/s relative to the platform, also in the anticlockwise direction. The new angular speed of the platform in (rad/s) is (take (I=0.02 kg-m2))

Solution

Given,

Moment of inertia of platform (disc) I=0.02 kg-m2

Radius of platform (disc) R=1 m

Speed of man w.r.t. platform vmp=1 m/s

Mass of man M=50 kg

Angular speed initially, ω=20 rad/s

Initially:

Initial angular momentum about central axis

Li=(I+MR2)ω

Finally:

Final angular momentum about central axis

Lf=Iω′+MvR

Lf=(I+MR2)ω′+MvR

Here,

(I+MR2) is the MOI of the system (man+platform)

ω′= angular velocity when man starts moving with velocity v w.r.t platform

Here, disc and man both are moving in anti-clockwise direction. Hence according to right hand thumb rule, direction of angular momentum for both is same (upward axial direction)

As we can see, there is no external torque on the system, so the angular momentum of the system about the rotating axis remains conserved or same.

Hence Li=Lf

(I+MR2)ω=(I+MR2)ω′+MvR

ω′=(ω−MvRI+MR2)

ω′=[20(rad/s)−(50 kg)(1 m/s)(1 m)0.02( kgm2)+50(kg)×(1 m)2]

ω′=[20−5050.02]

ω′=19 rad/s

Moment of inertia of platform (disc) I=0.02 kg-m2

Radius of platform (disc) R=1 m

Speed of man w.r.t. platform vmp=1 m/s

Mass of man M=50 kg

Angular speed initially, ω=20 rad/s

Initially:

Initial angular momentum about central axis

Li=(I+MR2)ω

Finally:

Final angular momentum about central axis

Lf=Iω′+MvR

Lf=(I+MR2)ω′+MvR

Here,

(I+MR2) is the MOI of the system (man+platform)

ω′= angular velocity when man starts moving with velocity v w.r.t platform

Here, disc and man both are moving in anti-clockwise direction. Hence according to right hand thumb rule, direction of angular momentum for both is same (upward axial direction)

As we can see, there is no external torque on the system, so the angular momentum of the system about the rotating axis remains conserved or same.

Hence Li=Lf

(I+MR2)ω=(I+MR2)ω′+MvR

ω′=(ω−MvRI+MR2)

ω′=[20(rad/s)−(50 kg)(1 m/s)(1 m)0.02( kgm2)+50(kg)×(1 m)2]

ω′=[20−5050.02]

ω′=19 rad/s

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