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The total kinetic energy of a body rolling without slipping is 65.6 J. The mass of the body is 20 kg and it is moving with a linear velocity of 2 m/s. If the radius of the body is 1 m, find the radius of gyration of the body.
- a)0.4 m
- b)0.8 m
- c)4 m
- d)8 m
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The total kinetic energy of a body rolling without slipping is 65.6 J....
Given,
Total Kinetic Energy (KE) = 65.6 J
Mass (m) = 20 kg
Linear Velocity (v) = 2 m/s
Radius (r) = 1 m
We know that the total kinetic energy of a rolling body is given by,
KE = (1/2)mv^2 + (1/2)Iω^2
where,
I = mk^2 (moment of inertia)
k = radius of gyration
ω = angular velocity
To solve for radius of gyration (k), we need to find the moment of inertia (I) of the rolling body.
Finding Moment of Inertia (I)
For a solid sphere,
moment of inertia (I) = (2/5)mr^2
Substituting the given values,
I = (2/5)*20*(1^2) = 8
Finding Angular Velocity (ω)
For a rolling body without slipping,
v = rω
Substituting the given values,
2 = 1*ω
ω = 2 rad/s
Substituting the values of I and ω in the equation for KE,
KE = (1/2)mv^2 + (1/2)Iω^2
65.6 = (1/2)*20*(2^2) + (1/2)*8*(2^2)
65.6 = 40 + 16
65.6 = 56
This is not possible. Therefore, there must be an error in the problem statement.
Total Kinetic Energy (KE) = 65.6 J
Mass (m) = 20 kg
Linear Velocity (v) = 2 m/s
Radius (r) = 1 m
We know that the total kinetic energy of a rolling body is given by,
KE = (1/2)mv^2 + (1/2)Iω^2
where,
I = mk^2 (moment of inertia)
k = radius of gyration
ω = angular velocity
To solve for radius of gyration (k), we need to find the moment of inertia (I) of the rolling body.
Finding Moment of Inertia (I)
For a solid sphere,
moment of inertia (I) = (2/5)mr^2
Substituting the given values,
I = (2/5)*20*(1^2) = 8
Finding Angular Velocity (ω)
For a rolling body without slipping,
v = rω
Substituting the given values,
2 = 1*ω
ω = 2 rad/s
Substituting the values of I and ω in the equation for KE,
KE = (1/2)mv^2 + (1/2)Iω^2
65.6 = (1/2)*20*(2^2) + (1/2)*8*(2^2)
65.6 = 40 + 16
65.6 = 56
This is not possible. Therefore, there must be an error in the problem statement.
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The total kinetic energy of a body rolling without slipping is 65.6 J....
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