Advertisements
Advertisements
प्रश्न
A torsional pendulum consists of a solid disc connected to a thin wire (α = 2.4 × 10–5°C–1) at its centre. Find the percentage change in the time period between peak winter (5°C) and peak summer (45°C).
उत्तर
Given:
Coefficient of linear expansion of the wire, α = 2.4 × 10–5 °C–1
Let I0 be the moment of inertia of the torsional pendulum at 0 °C.
If K is the torsional constant of the wire, then time period of torsional pendulum (T):
`T = 2pi sqrt(1/K)`...(1)
Here, I = moment of inertia after change in temperature
When the temperature is changed by Δθ, moment of inertia (I),
l = l0(1+2αΔθ)
On substituting the value of I in equation(1), we get:
`T = 2 pi sqrt( I_0(1+2αΔθ))/sqrt K`
In winter , Δθ = 5 °C
∴ Time period , (T1)
= `2pi sqrt( I_0(1+2αΔθ))/sqrt K`
In summer , Δθ = 45 °C
Time period (T2)
`=2pi sqrt( I_0(1+2αΔθ))/sqrt K`
So,
`T_2/T_1 = sqrt(1+90α)/sqrt(1+10α)`
=`sqrt(1+90 xx 2.4 xx 10^-5)/sqrt(1+10 xx 2.4 xx 10^-5)`
`=> T_2/T_1 = sqrt(1.00216)/sqrt(1.00024)`
% change = `(T_2/T_1 -1) xx 100`
= 0.0959 %
⇒ % change in time period ≈ 9.6 × 10-2 %
Therefore, the percentage change in time period of a torsional pendulum between peak winters and peak summers is 9.6 × 10–2 % .
APPEARS IN
संबंधित प्रश्न
Two absolute scales A and B have triple points of water defined to be 200 A and 350 B. What is the relation between TA and TB?
Answer the following:
The triple-point of water is a standard fixed point in modern thermometry. Why? What is wrong in taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale)?
Consider the following statements.
(A) The coefficient of linear expansion has dimension K–1.
(B) The coefficient of volume expansion has dimension K–1.
The steam point and the ice point of a mercury thermometer are marked as 80° and 20°. What will be the temperature on a centigrade mercury scale when this thermometer reads 32°?
The pressure measured by a constant volume gas thermometer is 40 kPa at the triple point of water. What will be the pressure measured at the boiling point of water (100°C)?
The pressures of the gas in a constant volume gas thermometer are 80 cm, 90 cm and 100 cm of mercury at the ice point, the steam point and in a heated wax bath, respectively. Find the temperature of the wax bath.
Four 2 cm × 2 cm × 2 cm cubes of ice are taken out from a refrigerator and are put in 200 ml of a drink at 10°C. (a) Find the temperature of the drink when thermal equilibrium is attained in it. (b) If the ice cubes do not melt completely, find the amount melted. Assume that no heat is lost to the outside of the drink and that the container has negligible heat capacity. Density of ice = 900 kg m−3, density of the drink = 1000 kg m−3, specific heat capacity of the drink = 4200 J kg−1 K−1, latent heat of fusion of ice = 3.4 × 105 J kg−1.
Two metre scales, one of steel and the other of aluminium, agree at 20°C. Calculate the ratio aluminium-centimetre/steel-centimetre at (a) 0°C, (b) 40°C and (c) 100°C. α for steel = 1.1 × 10–5 °C–1 and for aluminium = 2.3 × 10–5°C–1.
A metre scale made of steel reads accurately at 20°C. In a sensitive experiment, distances accurate up to 0.055 mm in 1 m are required. Find the range of temperature in which the experiment can be performed with this metre scale. Coefficient of linear expansion of steel = 11 × 10–6 °C–1.
An aluminium can of cylindrical shape contains 500 cm3 of water. The area of the inner cross section of the can is 125 cm2. All measurements refer to 10°C.
Find the rise in the water level if the temperature increases to 80°C. The coefficient of linear expansion of aluminium is 23 × 10–6 °C–1 and the average coefficient of the volume expansion of water is 3.2 × 10–4 °C–1.
A glass vessel measures exactly 10 cm × 10 cm × 10 cm at 0°C. It is filled completely with mercury at this temperature. When the temperature is raised to 10°C, 1.6 cm3 of mercury overflows. Calculate the coefficient of volume expansion of mercury. Coefficient of linear expansion of glass = 6.5 × 10–1 °C–1.
A cube of iron (density = 8000 kg m−3, specific heat capacity = 470 J kg−1 K−1) is heated to a high temperature and is placed on a large block of ice at 0°C. The cube melts the ice below it, displaces the water and sinks. In the final equilibrium position, its upper surface just goes inside the ice. Calculate the initial temperature of the cube. Neglect any loss of heat outside the ice and the cube. The density of ice = 900 kg m−3 and the latent heat of fusion of ice = 3.36 × 105 J kg−1.
A steel rod is rigidly clamped at its two ends. The rod is under zero tension at 20°C. If the temperature rises to 100°C, what force will the rod exert on one of the clamps? Area of cross-section of the rod is 2.00 mm2. Coefficient of linear expansion of steel is 12.0 × 10–6 °C–1 and Young's modulus of steel is 2.00 × 1011 Nm–2.
A circular disc made of iron is rotated about its axis at a constant velocity ω. Calculate the percentage change in the linear speed of a particle of the rim as the disc is slowly heated from 20°C to 50°C, keeping the angular velocity constant. Coefficient of linear expansion of iron = 1.2 × 10–5 °C–1.
Solve the following problem.
In a random temperature scale X, water boils at 200 °X and freezes at 20 °X. Find the boiling point of a liquid in this scale if it boils at 62 °C.
If the temperature on the Fahrenheit scale is 140 °F, then the same temperature on the Kelvin scale will be:
The graph between two temperature scales A and B is shown in figure. Between upper fixed point and lower fixed point there are 150 equal division on scale A and 100 on scale B. The relationship for conversion between the two scales is given by ______.
