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The quantities of heat required to raise the temperature of two solid copper spheres of radii r1 and r2 (r1 = 1.5 r2) through 1 K are in the ratio :
- a)3 / 2
- b)5 / 3
- c)27 / 8
- d)9 / 4
Correct answer is option 'C'. Can you explain this answer?
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The quantities of heat required to raise the temperature of two solid ...
Q = msΔT s is same as material is same
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The quantities of heat required to raise the temperature of two solid ...
Given: Radii of two solid copper spheres are r1 and r2 (r1=1.5r2)
To find: Ratio of quantities of heat required to raise their temperature by 1 K
Solution:
Let the masses of the two copper spheres be m1 and m2 respectively.
The volume of a sphere is given by V = (4/3)πr^3
Thus, the volumes of the two spheres are V1 = (4/3)πr1^3 and V2 = (4/3)πr2^3
Since copper is a homogeneous material, we can assume that the densities of the two spheres are the same.
Therefore, their masses are given by m1 = ρV1 and m2 = ρV2, where ρ is the density of copper.
Assuming that the specific heat capacity of copper is the same for both spheres, the quantities of heat required to raise their temperatures by 1 K are given by:
Q1 = m1cΔT and Q2 = m2cΔT,
where c is the specific heat capacity of copper and ΔT = 1 K.
Substituting the expressions for m1 and m2, we get:
Q1 = ρV1cΔT and Q2 = ρV2cΔT
Dividing Q1 by Q2, we get:
Q1/Q2 = (ρV1cΔT)/(ρV2cΔT) = V1/V2 = (r1^3)/(r2^3)
Given that r1 = 1.5r2, we can substitute this in the above equation to get:
Q1/Q2 = (1.5^3)/(1^3) = 27/8
Therefore, the required ratio is 27/8.
Hence, the correct option is (C) 27/8.
To find: Ratio of quantities of heat required to raise their temperature by 1 K
Solution:
Let the masses of the two copper spheres be m1 and m2 respectively.
The volume of a sphere is given by V = (4/3)πr^3
Thus, the volumes of the two spheres are V1 = (4/3)πr1^3 and V2 = (4/3)πr2^3
Since copper is a homogeneous material, we can assume that the densities of the two spheres are the same.
Therefore, their masses are given by m1 = ρV1 and m2 = ρV2, where ρ is the density of copper.
Assuming that the specific heat capacity of copper is the same for both spheres, the quantities of heat required to raise their temperatures by 1 K are given by:
Q1 = m1cΔT and Q2 = m2cΔT,
where c is the specific heat capacity of copper and ΔT = 1 K.
Substituting the expressions for m1 and m2, we get:
Q1 = ρV1cΔT and Q2 = ρV2cΔT
Dividing Q1 by Q2, we get:
Q1/Q2 = (ρV1cΔT)/(ρV2cΔT) = V1/V2 = (r1^3)/(r2^3)
Given that r1 = 1.5r2, we can substitute this in the above equation to get:
Q1/Q2 = (1.5^3)/(1^3) = 27/8
Therefore, the required ratio is 27/8.
Hence, the correct option is (C) 27/8.
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