Advertisements
Advertisements
प्रश्न
Two steel rods and an aluminium rod of equal length l0 and equal cross-section are joined rigidly at their ends, as shown in the figure below. All the rods are in a state of zero tension at 0°C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are αa and αs, respectively. Young's modulus of aluminium is Ya and of steel is Ys.
| Steel |
| Aluminium |
| Steel |
उत्तर
Given:-
Initial length of the system (two identical steel rods + an aluminium rod) at 0 °C = l0
Coefficient of linear expansion of steel and aluminium are αs and αAl, respectively.
Temperature is raised by θ.
So, the change in temperature, Δθ = θ - 0 ° C = θ
Young's modulus of steel and aluminium are γs and γAl, respectively.
If l be the final length of the system at temperature θ,
strain on the system = `(l-l_0)/l_0` ...(1)
Young's modulus = `("Stress") /("Strain")`
Therefore, the total strain on the system = `("Total stress on the system")/("Total Young's modulus of the system")`
Now, total stress = stress due to the two steel rods + stress due to the aluminium rod
`"Stress" = "F"/"A" = αY Δθ`
`"Total stress" = ϒ_sa_stheta + gamma_s a_stheta+ϒ_(Al)a_(Al)theta`
`=2gamma_sa_stheta +gamma_(Al)a_(Al)theta` ...(2)
Young's modulus of the system,
Y = ϒs +ϒs+ϒAl = 2ϒs + ϒAl ...(3)
Using (1), (2) and (3), we get:
Strain on the system = `(2ϒ_sα_sθ + ϒ_(Al)α_Alθ)/(2ϒ_s+γ_Al)`
`=> (l - l_0)/l = (2ϒ_sα_sθ + ϒ_(Al)α_Alθ)/(2ϒ_s+γ_Al)`
`l =l_0[ 1+ (2ϒ_sα_sθ + ϒ_(Al)α_Alθ)/(2ϒ_s+γ_Al)]`
Therefore, the final length of the system will be l0 [1+`(2ϒ_sα_sθ + ϒ_(Al)α_Alθ)/(2ϒ_s+γ_Al)` ] , where l0 is its initial length.
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?
The electrical resistance in ohms of a certain thermometer varies with temperature according to the approximate law:
R = Ro [1 + α (T – To)]
The resistance is 101.6 Ω at the triple-point of water 273.16 K, and 165.5 Ω at the normal melting point of lead (600.5 K). What is the temperature when the resistance is 123.4 Ω?
Two ideal gas thermometers Aand Buse oxygen and hydrogen respectively. The following observations are made:
| Temperature | Pressure thermometer A | Pressure thermometer B |
| Triple-point of water | 1.250 × 105 Pa | 0.200 × 105 Pa |
| Normal melting point of sulphur | 1.797 × 105 Pa | 0.287 × 105 Pa |
(a) What is the absolute temperature of the normal melting point of sulphur as read by thermometers Aand B?
(b) What do you think is the reason behind the slight difference in answers of thermometers Aand B? (The thermometers are not faulty). What further procedure is needed in the experiment to reduce the discrepancy between the two readings?
In defining the ideal gas temperature scale, it is assumed that the pressure of the gas at constant volume is proportional to the temperature T. How can we verify whether this is true or not? Do we have to apply the kinetic theory of gases? Do we have to depend on experimental result that the pressure is proportional to temperature?
If the temperature of a uniform rod is slightly increased by ∆t, its moment of inertia I about a perpendicular bisector increases by
In which of the following pairs of temperature scales, the size of a degree is identical?
(a) Mercury scale and ideal gas scale
(b) Celsius scale and mercury scale
(c) Celsius scale and ideal gas scale
(d) Ideal gas scale and absolute scale
A constant-volume thermometer registers a pressure of 1.500 × 104 Pa at the triple point of water and a pressure of 2.050 × 104 Pa at the normal boiling point. What is the temperature at the normal boiling point?
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 pressure of the gas in a constant volume gas thermometer is 70 kPa at the ice point. Find the pressure at the steam point.
An aluminium vessel of mass 0.5 kg contains 0.2 kg of water at 20°C. A block of iron of mass 0.2 kg at 100°C is gently put into the water. Find the equilibrium temperature of the mixture. Specific heat capacities of aluminium, iron and water are 910 J kg−1 K−1, 470 J kg−1 K−1 and 4200 J kg−1 K−1 respectively.
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 ball is dropped on a floor from a height of 2.0 m. After the collision it rises up to a height of 1.5 m. Assume that 40% of the mechanical energy lost goes as thermal energy into the ball. Calculate the rise in the temperature of the ball in the collision. Heat capacity of the ball is 800 J K−1.
A copper cube of mass 200 g slides down on a rough inclined plane of inclination 37° at a constant speed. Assume that any loss in mechanical energy goes into the copper block as thermal energy. Find the increase in the temperature of the block as it slides down through 60 cm. Specific heat capacity of copper = 420 J kg−1 K−1.
A metal block of density 600 kg m−3 and mass 1.2 kg is suspended through a spring of spring constant 200 N m−1. The spring-block system is dipped in water kept in a vessel. The water has a mass of 260 g and the bloc is at a height 40 cm above the bottom of the vessel. If the support of the spring is broken, what will be the rise in the temperature of the water. Specific heat capacity of the block is 250 J kg−3 K−1 and that of water is 4200 J kg−1 K−1. Heat capacities of the vessel and the spring are negligible.
Answer the following question.
How a thermometer is calibrated?
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.
Calculate the temperature which has same numeral value on celsius and Fahrenheit scale.
