Concept explainers
(a)
Prove that the kinetic energy can be written as,
(a)
Answer to Problem 10.33P
it is proved that the kinetic energy can be written as,
Explanation of Solution
Let the mass of each particle of the rigid body is
Write the expression for the
Here,
Write the expression for the kinetic energy.
Here,
Use equation (I) in (II),
The expansion of the vector
Let
Here,
Apply equation (IV) in (III),
The total kinetic energy of the rigid body is,
Conclusion:
Therefore, it is proved that the kinetic energy can be written as,
(b)
Prove that the
(b)
Answer to Problem 10.33P
It is proved that the angular momentum
Explanation of Solution
Write the expression for the angular momentum of each particle.
Here,
The momentum of the particle is given by,
Use equation (I) in (VI),
Use equation (VII) in (V),
According to the vector identity,
Using equation (VIII)
Conclusion:
The total angular momentum of the rigid body is
Therefore, It is proved that the angular momentum
(c)
Using results from part (a) and (b), prove that
(c)
Answer to Problem 10.33P
Using results from part (a) and (b), it is proved that
Explanation of Solution
In order to prove
Thus it is proved that
The expression for the angular momentum is,
Using this, the above equation becomes,
Here,
But the matrix form of
Here,
Conclusion:
Using results from part (a) and (b), it is proved that
(d)
Show that with respect to the principal axes,
(d)
Answer to Problem 10.33P
It is Shown that with respect to the principal axes,
Explanation of Solution
Write the expression for the moment of inertia tensor
Generally, moment of inertia tensor can be written as,
Then the
Then
Conclusion:
It is Shown that with respect to the principal axes,
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Chapter 10 Solutions
Classical Mechanics
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