Problem 11 - Applied Physics

Question

Find the cost of operating a \(1.50-\mathrm{A}\) motor on a \(11 \overline{0}-\mathrm{V}\) circuit for \(2.00 \mathrm{~h}\) at \(\$ 0.08 / \mathrm{kWh}\).

Step-by-Step Solution

Answer

The cost is $0.0264.

Step 1: Calculate Power

The power of the motor is calculated using the formula: \[ P = IV \]where \( P \) is the power in watts, \( I \) is the current in amperes, and \( V \) is the voltage in volts. Plug in the given values:\[ P = 1.50 \, \text{A} \times 110 \, \text{V} = 165 \, \text{W} \].

Step 2: Convert Power to Kilowatts

Since electricity cost is usually given in kilowatt-hours, we need to convert the power from watts to kilowatts. This is done by dividing by 1000:\[ P_{\text{kW}} = \frac{165 \, \text{W}}{1000} = 0.165 \, \text{kW} \].

Step 3: Calculate Energy Used

Energy used (in kilowatt-hours) is calculated by multiplying the power in kilowatts by the time in hours:\[ E = P_{\text{kW}} \times t \]\[ E = 0.165 \, \text{kW} \times 2.00 \, \text{h} = 0.33 \, \text{kWh} \].

Step 4: Find Total Cost

To find the total cost of operating the motor, multiply the energy used by the cost per kilowatt-hour:\[ \text{Cost} = E \times \text{Cost per kWh} \]\[ \text{Cost} = 0.33 \, \text{kWh} \times 0.08 \, \text{\\(/kWh} = 0.0264 \, \text{\\)} \].

Key Concepts

Power Calculation

Calculating power is the first step in understanding how much electrical energy an appliance uses. Power is the rate at which energy is used or produced, and it is calculated using the simple formula: \[ P = I \times V \]Here:

• \( P \) is the power in watts (W).
• \( I \) is the current in amperes (A).
• \( V \) is the voltage in volts (V).

Plugging in the values for our motor, which has a current of 1.50 A and operates on a 110 V circuit, gives us:\[ P = 1.50 \, \text{A} \times 110 \, \text{V} = 165 \, \text{W} \] This calculation tells us that the motor consumes 165 watts of electrical power.

Energy Consumption

Energy consumption refers to the amount of energy used over time. For electricity, we typically measure energy in kilowatt-hours (kWh), which combines power in kilowatts with time in hours. To calculate energy usage, the formula is:\[ E = P_{\text{kW}} \times t \]Where:

• \( E \) is the energy in kilowatt-hours.
• \( P_{\text{kW}} \) is the power in kilowatts.
• \( t \) is the time in hours.

For our motor, we first converted its power from watts to kilowatts: \[ P_{\text{kW}} = \frac{165 \, \text{W}}{1000} = 0.165 \, \text{kW} \] Then, using this power for 2 hours:\[ E = 0.165 \, \text{kW} \times 2.00 \, \text{h} = 0.33 \, \text{kWh} \]This indicates the motor consumes 0.33 kilowatt-hours of energy.

Unit Conversion

Unit conversion is crucial for solving problems involving electricity costs. Calculating electricity consumption costs requires converting from watts (W) to kilowatts (kW) because electricity costs are typically expressed in terms of kilowatt-hours (kWh).To convert watts to kilowatts, remember that:\[ 1 \text{ kW} = 1000 \, \text{W} \]Thus, to convert from watts to kilowatts, divide by 1000:\[ P_{\text{kW}} = \frac{P \, \text{W}}{1000} \]In our exercise, the power of the motor is 165 W. Converting this to kilowatts:\[ P_{\text{kW}} = \frac{165 \, \text{W}}{1000} = 0.165 \, \text{kW} \]This step is vital for accurately calculating energy consumption and costs.

Cost Analysis

Cost analysis helps us determine how much it costs to run an electrical appliance. By knowing the energy consumption and the cost per kilowatt-hour (kWh), we can calculate the total cost.The formula for determining the total electricity cost is:\[ \text{Cost} = E \times \text{Cost per kWh} \]Where:

• \( E \) is the energy consumed in kilowatt-hours.
• Cost per kWh is the price of electricity per kilowatt-hour.

In our scenario, we have:

• Energy used = 0.33 kWh
• Cost per kWh = \(0.08

Thus, the total cost is:\[ \text{Cost} = 0.33 \, \text{kWh} \times 0.08 \, \text{\\)}/\text{kWh} = 0.0264 \, \text{\\(} \]Therefore, it costs \( \\)0.0264 \) to operate the motor for 2 hours.