Question

Consider two bodies A and B with respective
angular speeds 2 rad/s and 4 rad/s. If they have
same mass and same rotational kinetic energy
then the ratio of radius of gyration is
2:1
1:3
5:1
2:3​

Answer 1

Answer:

Given:

Angular speed of body A = 2 rad/s

Angular speed of body B = 4 rad/s

To find:

The ratio of radius of gyration of the two bodies.

Solution:

We know that radius of gyration

k = \sqrt{ \frac{i}{m} } k=

m

i

where m is the mass of the body and I is the moment of inertia.

We are given that they have same rotational kinetic energy.

And kinetic energy is given by

\frac{1}{2} i {w}^{2}

2

1

iw

2

where w is the angular velocity.

So,

\frac{1}{2} i1 {w1}^{2} = \frac{1}{2} i2 {w}^{2}

2

1

i1w1

2

=

2

1

i2w

2

\frac{1}{2} i1 ({2}^{2}) = \frac{1}{2}i2 ({4}^{2} )

2

1

i1(2

2

)=

2

1

i2(4

2

)

i1 = 4i2i1=4i2

Ratio of radius of gyration will be:

\frac{k1}{k2} = \sqrt{ \frac{i1}{i2} }

k2

k1

=

i2

i1

\frac{k1}{k2} = \sqrt{ \frac{4i2}{i2} }

k2

k1

=

i2

4i2

k1/ k2 = 2/1

Therefore the ratio of radius of gyration is 2:1.