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.
Advertisements
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.
The coefficient of volume expansion of glycerin is 49 × 10–5 K–1. What is the fractional change in its density for a 30 °C rise in temperature?
If an automobile engine is overheated, it is cooled by pouring water on it. It is advised that the water should be poured slowly with the engine running. Explain the reason.
A system X is neither in thermal equilibrium with Y nor with Z. The systems Y and Z
A gas thermometer measures the temperature from the variation of pressure of a sample of gas. If the pressure measured at the melting point of lead is 2.20 times the pressure measured at the triple point of water, find the melting point of lead.
Answer the following question.
Derive the relation between three coefficients of thermal expansion.
Answer the following question.
State applications of thermal expansion.
A metre scale made of a metal reads accurately at 25 °C. Suppose in an experiment an accuracy of 0.12 mm in 1 m is required, the range of temperature in which the experiment can be performed with this metre scale is ______.(coefficient of linear expansion of the metal is `20 xx 10^-6 / (°"C")`
A metal rod of cross-sectional area 3 × 10-6 m2 is suspended vertically from one end has a length 0.4 m at 100°C. Now the rod is cooled upto 0°C, but prevented from contracting by attaching a mass 'm' at the lower end. The value of 'm' is ______.
(Y = 1011 N/m2, coefficient of linear expansion = 10-5/K, g = 10m/s2)
A metal rod of length Land cross-sectional area A is heated through T °C. What is the force required to prevent the expansion of the rod lengthwise?
(Y = Young's modulus of material of the rod, α = coefficient of linear expansion of the rod.)
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly ______.
At what temperature a gold ring of diameter 6.230 cm be heated so that it can be fitted on a wooden bangle of diameter 6.241 cm? Both diameters have been measured at room temperature (27°C). (Given: coefficient of linear thermal expansion of gold αL = 1.4 × 10-5 K-1).
A solid metallic cube having a total surface area of 24 m2 is uniformly heated. If its temperature is increased by 10°C, calculate the increase in the volume of the cube.
(Given: α = 5.0 × 10−4°C−1)
A metal rod Y = 2 × 1012 dyne cm-2 of coefficient of linear expansion 1.6 × 10-5 per °C has its temperature raised by 20°C. The linear compressive stress to prevent the expansion of the rod is ______.
When heat is given to a substance, it generally ______.
The increase in the dimensions of a body due to an increase in its temperature is called ______.
