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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
Prove the theorem of perpendicular axes.
(Hint: Square of the distance of a point (x, y) in the x–y plane from an axis through the origin perpendicular to the plane is x2 + y2).
Find the radius of gyration of circular ring of radius r about a line perpendicular to the plane of the ring and passing through one of its particles.
Answer in brief:
State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.
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(g = acceleration due to gravity)
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A metal ring has a moment of inertia 2 kg·m2 about a transverse axis through its centre. It is melted and recast into a thin uniform disc of the same radius. What will be the disc's moment of inertia about its diameter?
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.
Prove the theorem of perpendicular axes about the moment of inertia.
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A thin rod of length ‘4L’ and mass ‘4m’ is bent at the points as shown in the figure. The moment of inertia of the rod about an axis passing through point ‘O’ and perpendicular to plane of the paper is:
