Assume that a certain city consumes electrical energy at an average rate of 2.0× 10^9 W. What would be the mass change
in producing enough energy to keep this city running for 21 weeks?
E = m c^2
so
m = E/c^2
E = 2*10^9 *21 * 7 *24 *3600 = 2.54*10^16
c = 3*10^8 m/s
c^2 = 9 * 10^16
so
m = 2.54 / 9 = .282 Kg = 282 grams
To find the mass change in producing enough energy to keep the city running for 21 weeks, we need to calculate the total energy consumed by the city during this time period.
First, we can calculate the energy consumed per week by multiplying the average power consumption rate by the number of seconds in a week:
Energy per week = Power consumption rate × Time per week
Time per week = 7 days/week × 24 hours/day × 60 minutes/hour × 60 seconds/minute
Substituting the given values:
Time per week = 7 × 24 × 60 × 60 seconds/week
Next, we can calculate the total energy consumed over 21 weeks by multiplying the energy consumed per week by the number of weeks:
Total energy consumed = Energy per week × Number of weeks
Now, we can find the required mass change by using the equation E = mc², where E is the energy and c is the speed of light. Rearranging the equation to solve for mass, we have:
m = E / c²
Let's calculate step by step:
Average power consumption rate = 2.0 × 10^9 W
Number of weeks = 21
Time per week = 7 × 24 × 60 × 60 seconds/week
Energy per week = Average power consumption rate × Time per week
Total energy consumed = Energy per week × Number of weeks
The speed of light, c, is approximately 3.0 × 10^8 m/s.
Plugging in the values:
m = Total energy consumed / (c²)
Now we can calculate the mass change.
To calculate the mass change in producing enough energy to keep the city running for 21 weeks, you need to convert the energy consumption into a mass measure using Einstein's mass-energy equivalence formula, E = mc^2.
Step 1: Convert energy consumption into joules (J)
Given the average energy consumption rate of the city, which is 2.0 × 10^9 W (watts), we can multiply it by the number of seconds in a week to get the total energy consumption in joules (J).
Total energy consumption = 2.0 × 10^9 W × 7 days/week × 24 hours/day × 3600 seconds/hour
Step 2: Convert joules (J) to kilograms (kg)
Using Einstein's mass-energy equivalence formula, E = mc^2, we can rearrange it to solve for mass (m).
m = E / c^2
Where:
E is the energy in joules (J)
c is the speed of light (approximately 3.0 × 10^8 m/s)
Substituting the value of the total energy consumption in joules (J) and the speed of light (c), we can calculate the mass (m).
Step 3: Calculate the mass change
Now that we have the mass (m) value, we can calculate the mass change by considering the energy consumption for 21 weeks. Multiply the mass obtained in step 2 by 21.
Mass change = Mass × 21
Let's calculate the mass change using these steps:
Step 1:
Total energy consumption = 2.0 × 10^9 W × 7 days/week × 24 hours/day × 3600 seconds/hour
Total energy consumption ≈ 2.2656 × 10^17 J
Step 2:
m = 2.2656 × 10^17 J / (3.0 × 10^8 m/s)^2
m ≈ 2.517 × 10^9 kg
Step 3:
Mass change = 2.517 × 10^9 kg × 21
Mass change ≈ 5.286 × 10^10 kg
Therefore, the mass change in producing enough energy to keep the city running for 21 weeks would be approximately 5.286 × 10^10 kilograms.