Q34 of 67 Page 98

A rod of negligible heat capacity has length 20 cm, area of cross-section 1.0 cm2 and thermal conductivity 200 W m–1 °C–1. The temperature of one end is maintained at 0°C and that of the other end is slowly and linearly varied from 0°C to 60°C in 10 minutes. Assuming no loss of heat through the sides, find the total heat transmitted through the rod in these 10 minutes.

Given-
Length of the rod, l = 20 cm = 0.2 m
Area of cross section of the rod, A = 1.0 cm2 = 1.0×10-4 m2


Thermal conductivity of the rod, k = 200 W m-1°C-1


Also, the temperature of one end is maintained at 0°C and that of the other end is varied from 0°C to 60°C in 10 minutes.


Hence rate of increase of the temperature at one end is 0.1°C per second.


⇒ = = 0.1°C/s


Now, rate of flow of heat –


=


= +...........+ (


= )


We know Arithmetic Progressions –


For a series given by


a+2a+...................na


Sum of n terms is given by


Sum= , where a is the first term, d is the difference between first and second term and n is the number of terms.


Substituting the values, for 10 minutes –


Here a= 0,1. d = 0.1 and n = 60 for 10mins,



=


= 1800J


Hence , the total heat transmitted through the rod in these 10 minutes is 1800J


More from this chapter

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32

Seven rods A, B, C, D, E, F and G are joined as shown in figure. All the rods have equal cross-sectional area A and length ℓ. The thermal conductivities of the rods are KA = KC = KD, KD = 2KD, KE = 3KD, KF = 4KD and KD = 5KD. The rod E is kept at a constant temperature T1 and the rod G is kept at a constant temperature T2(T2> T1).

(a) Show that the rod F ahs a uniform temperature T = (T1 + 2T2)/3.


(b) Find the rate of heat flowing from the source which maintains the temperature T2.



 33

Find the rate of heat flow through a cross-section of the rod shown in figure (θ2> θ1). Thermal conductivity of the material of the rod is K.


 35

A hollow metallic sphere of radius 20 cm surrounds a concentric metallic sphere of radius 5 cm. The space between the two spheres in filled with a nonmetallic material. The inner and outer spheres are maintained at 50°C and 10°C respectively and it is found that 100 J of heat passes from the inner sphere to the outer sphere per second. Find the thermal conductivity of the material between the spheres.

 36

Figure shows two adiabatic vessels, each containing a mass m of water at different temperatures. The ends of a metal rod of length L, area of cross-section A and thermal conductivity K, are inserted in the water as shown in the figure. Find the time taken for the difference between the temperatures in the vessels to become half of the original value. The specific heat capacity of water is s. Neglect the heat capacity of the rod and the container and any loss of heat to the atmosphere.