You've built a device that uses the energy from a rapidly discharged capacitor to launch the capacitor straight up. One capacitor, with a mass of 3.5 g, is launched to a height of 1.6 m after having been charged to 100 V. What is its capacitance in μF?

Verified step by step guidance

1. Convert the mass of the capacitor from grams to kilograms for consistency in SI units. Use the conversion: \( 1 \ \text{g} = 0.001 \ \text{kg} \). Thus, \( m = 3.5 \ \text{g} = 0.0035 \ \text{kg} \).
2. Determine the potential energy of the capacitor at its maximum height. At the peak, all the energy is gravitational potential energy, given by \( U = m g h \), where \( g = 9.8 \ \text{m/s}^2 \) is the acceleration due to gravity, \( m \) is the mass, and \( h \) is the height.
3. Relate the gravitational potential energy to the energy stored in the capacitor. The energy stored in a charged capacitor is given by \( E = \frac{1}{2} C V^2 \), where \( C \) is the capacitance and \( V \) is the voltage. Set \( \frac{1}{2} C V^2 = m g h \) to equate the two energies.
4. Rearrange the equation \( \frac{1}{2} C V^2 = m g h \) to solve for the capacitance \( C \). This gives \( C = \frac{2 m g h}{V^2} \).
5. Substitute the known values into the equation: \( m = 0.0035 \ \text{kg} \), \( g = 9.8 \ \text{m/s}^2 \), \( h = 1.6 \ \text{m} \), and \( V = 100 \ \text{V} \). Perform the calculation to find \( C \) in farads, then convert the result to microfarads (\( 1 \ \mu\text{F} = 10^{-6} \ \text{F} \)).

Key Concepts

Capacitance

Capacitance is a measure of a capacitor's ability to store electrical energy. It is defined as the amount of charge stored per unit voltage, typically expressed in farads (F). In this context, capacitance is crucial for determining how much energy the capacitor can store and subsequently release to launch the device.

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. For an object of mass m raised to a height h, the potential energy is given by the formula PE = mgh, where g is the acceleration due to gravity. This concept is essential for calculating the energy transferred from the capacitor to the mass during the launch.

Energy Stored in a Capacitor

The energy (E) stored in a capacitor is given by the formula E = 0.5 * C * V^2, where C is the capacitance and V is the voltage. This relationship is vital for solving the problem, as it allows us to relate the energy stored in the capacitor to the potential energy of the launched mass, enabling the calculation of the capacitance needed for the given height.