To tackle this problem, we need to break it down into several parts, focusing on the principles of heat transfer and Newton's Law of Cooling. Let's go through each part step by step.
Finding the Power of the Heater
First, we need to determine the power of the heater when it is switched on. The power can be calculated using the formula:
Power (P) = Heat Capacity (C) × Rate of Temperature Change (dT/dt)
Given that the heat capacity (C) of the metal block is 80 J/°C and the rate of temperature rise (dT/dt) is 2°C/s, we can substitute these values into the formula:
P = 80 J/°C × 2°C/s = 160 W
Calculating the Power Radiated Just After the Heater is Switched Off
Next, we need to find the power radiated by the block immediately after the heater is turned off. According to Newton's Law of Cooling, the power radiated can be expressed as:
P_radiated = k × (T_block - T_room)
Where:
- T_block is the temperature of the block (30°C just after the heater is switched off).
- T_room is the room temperature (20°C).
- k is a constant that relates to the cooling rate.
We know the rate of temperature fall is 0.2°C/s. This can also be expressed in terms of power:
P_radiated = C × (dT/dt)
Substituting the values:
P_radiated = 80 J/°C × (-0.2°C/s) = -16 W
The negative sign indicates that this is a loss of power (heat being radiated away).
Power Radiated at 25°C
Now, let's find the power radiated when the temperature of the block is 25°C. Using the same formula:
P_radiated = k × (T_block - T_room)
We need to determine the value of k. From the previous calculation, we can find k using the temperature difference at 30°C:
16 W = k × (30°C - 20°C) = k × 10°C
Solving for k gives:
k = 16 W / 10°C = 1.6 W/°C
Now, substituting back to find the power radiated at 25°C:
P_radiated = 1.6 W/°C × (25°C - 20°C) = 1.6 W/°C × 5°C = 8 W
Determining the Time the Heater was On
To find the time for which the heater was kept on, we can use the average power radiated during the heating process. We previously calculated the power radiated at 25°C as 8 W. Assuming this represents the average power during the heating process, we can set up the following equation:
Average Power = (Power of Heater - Average Power Radiated)
Substituting the known values:
8 W = (160 W - Average Power Radiated)
From this, we can find the average power radiated during the heating phase:
Average Power Radiated = 160 W - 8 W = 152 W
Now, we can find the time (t) the heater was on using the total energy supplied:
Energy = Power × Time
The energy supplied to raise the temperature from 20°C to 30°C (a 10°C increase) is:
Energy = C × ΔT = 80 J/°C × 10°C = 800 J
Now, substituting into the energy equation:
800 J = 160 W × t
Solving for t gives:
t = 800 J / 160 W = 5 seconds
Summary of Results
- Power of the heater: 160 W
- Power radiated just after switching off: 16 W
- Power radiated at 25°C: 8 W
- Time the heater was on: 5 seconds
This structured approach allows us to understand the heat transfer dynamics involved in this scenario, applying fundamental principles effectively. If you have any further questions or need clarification on any part, feel free to ask!
