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प्रश्न
Prove the theorem of parallel axes.
(Hint: If the centre of mass is chosen to be the origin `summ_ir_i = 0`
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उत्तर
Theorem of parallel axes: According to this theorem, moment of inertia of a rigid body about any axis AB is equal to moment of inertia of the body about another axis KL passing through centre of mass C of the body in a direction parallel to AB, plus the product of total mass M of the body and square of the perpendicular distance between the two parallel axes. If h is perpendicular distance between the axes AB and KL, then Suppose the rigid body is made up of n particles m1, m2, …. mn, mn at perpendicular distances r1, r2, ri…. rn. respectively from the axis KL passing through the centre of mass C of the body.
If h is the perpendicular distance of the particle of mass m{ from KL, then
The perpendicular distance of ith particular from the axis
`AB = (r_i + n)`
or `I_(AB) = sum_(i)m_i(r_i + h)^2`
`= sum_(i) m_ir_i^2 + sum_(i) m_ih^2 + 2h sum_(i) m_ir_i` ....(ii)
As the body is balanced about the centre of mass the algebraic sum of the moments of the weights of all particles about an axis passing through C must be zero
`sum_(i)(m_ig)r_i = 0 or g sum_(i) m_ir_i` or `sum_(i) m_ir_i = 0` ...(iii)
From equation (ii) we have
`I_(AB) = sum_(i) m_ir_i^2 + (summ_i)h^2 + 0`
or `I_(AB) = I_(KL) + Mh^2`
Where `I_"KL" = sum_(i) m_ir_i^2` and `M = sum m_i`
संबंधित प्रश्न
State the theorem of perpendicular axes about moment of inertia.
State an expression for the moment of intertia of a solid uniform disc, rotating about an axis passing through its centre, perpendicular to its plane. Hence derive an expression for the moment of inertia and radius of gyration:
i. about a tangent in the plane of the disc, and
ii. about a tangent perpendicular to the plane of the disc.
Prove the theorem of parallel axes about moment of inertia
State Brewster's law.
Answer in brief:
State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.
State and explain the theorem of parallel axes.
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(g = acceleration due to gravity)
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A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre 'O' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is:
Two particles A and B having equal charges are placed at a distance d apart. A third charged particle placed on the perpendicular bisection of AB at distance x. The third particle experiences maximum force when ______.
A uniform disc of mass 10 kg and radius 60 cm rotates about an axis perpendicular to its plane and passing through its centre at 1200 rpm. Calculate its rotation kinetic energy. [Take π2 = 10]
State and prove the theorem of the parallel axis about the moment of inertia.
