Advertisements
Advertisements
प्रश्न
Find out the increase in moment of inertia I of a uniform rod (coefficient of linear expansion α) about its perpendicular bisector when its temperature is slightly increased by ∆T.
उत्तर
Let the mass and length of a uniform rod be M and l respectively.
Moment of inertia of the rod about its perpendicular bisector. `(i) = (Ml^2)/12`
The increase in length of the rod when temperature is increased by ∆T is given by `∆l = l.α∆T` .....(i)
∴ New moment of inertia of the rod `(I) = M/12 (l + ∆l)^2`
= `M/12 (l^2 + ∆l^2 + 2I∆l)`
As the change in length ∆l is very small, therefore, neglecting `(∆l)^2`, we get
`I^' = M/12 (l^2 + 2l∆l)`
= `(Ml^2)/12 + (MI∆l)/6`
= `l + (MI∆l)/6`
∴ Increase in the moment of inertia `∆I = l - I`
= `(MI∆l)/6`
= `2 xx ((Ml^2)/12) (∆l)/l`
`∆I = 2*I α∆T` ......[Using equation (i)]
APPEARS IN
संबंधित प्रश्न
A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = 1.70 × 10–5 K–1.
A 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, assuming 50% of power is used up in heating the machine itself or lost to the surroundings Specific heat of aluminium = 0.91 J g–1 K–1
For a constant-volume gas thermometer, one should fill the gas at
Answer the following question.
State applications of thermal expansion.
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly ______.
An aluminium sphere is dipped into water. Which of the following is true?
The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is α. The sphere is heated a little by a temperature ∆T so that its new temperature is T + ∆T. The increase in the volume of the sphere is approximately ______.
Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of 57°C is drunk. You can take body (tooth) temperature to be 37°C and α = 1.7 × 10–5/°C, bulk modulus for copper = 140 × 109 N/m2.
Length of steel rod so that it is 5 cm longer than the copper rod at all temperatures should be ______ cm.
(α for copper = 1.7 × 10-5/°C and α for steel = 1.1 × 10-5/°C)
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
- Assertion A: When a rod lying freely is heated, no thermal stress is developed in it.
- Reason R: On heating, the length of the rod increases. In light of the above statements.
choose the correct answer from the options given below:
