Question

A metal sphere cools at a rate of \(2^{\circ }\text{C/min}\) when the temperature is \(80^{\circ }\text{C}\). If the temperature of the surroundings is \(30^{\circ }\text{C}\), then the rate of cooling when its temperature is \(50^{\circ }\text{C}\) will be

• A. \(0.4^{\circ }\text{C/min}\)
• B. \(2^{\circ }\text{C/min}\)
• C. \(1.2^{\circ }\text{C/min}\)
• D. \(0.8^{\circ }\text{C/min}\)

Hint:

The rate of cooling is proportional to the excess of the body's temperature over the surroundings. Compare the excess at the two temperatures.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
By Newton's law of cooling, the rate of cooling is proportional to \((T-T_{0})\), where \(T_{0}\) is the temperature of the surroundings.

Step 2: First state
At \(80^{\circ}\text{C}\) the excess is \(80-30=50^{\circ}\text{C}\) and the rate is \(2^{\circ}\text{C/min}\).

Step 3: Second state
At \(50^{\circ}\text{C}\) the excess is \(50-30=20^{\circ}\text{C}\).

Step 4: Calculation
\(\dfrac{R_{2}}{2}=\dfrac{20}{50}\), so \(R_{2}=0.8^{\circ}\text{C/min}\).

Final Answer:
The rate of cooling is 0.8 degrees Celsius per minute. \[ \boxed{\text{(D) }0.8^{\circ}\text{C/min}} \]