To solve the problem of how high the mass is projected when released from a compressed spring, we will follow these steps: ### Step 1: Understand the Problem We have a spiral spring with a force constant (k) of 500 N/m, and a mass (m) of 5 x 10^-3 kg (which is 0.005 kg). The spring is compressed by a distance (x) of 0.02 m (or 2 cm). When the spring is released, it will convert its potential energy into kinetic energy, which will then be converted into gravitational potential energy as the mass rises. ### Step 2: Calculate the Potential Energy Stored in the Spring The potential energy (PE) stored in a compressed spring can be calculated using the formula: \[ PE = \frac{1}{2} k x^2 \] Where: - \( k = 500 \, \text{N/m} \) (spring constant) - \( x = 0.02 \, \text{m} \) (compression of the spring) Substituting the values into the formula: \[ PE = \frac{1}{2} \times 500 \, \text{N/m} \times (0.02 \, \text{m})^2 \] Calculating \( (0.02)^2 \): \[ (0.02)^2 = 0.0004 \, \text{m}^2 \] Now substituting back into the equation: \[ PE = \frac{1}{2} \times 500 \times 0.0004 \] \[ PE = 250 \times 0.0004 = 0.1 \, \text{J} \] ### Step 3: Convert Potential Energy to Gravitational Potential Energy When the spring is released, the potential energy stored in the spring is converted into gravitational potential energy (GPE) as the mass rises. The gravitational potential energy can be calculated using the formula: \[ GPE = mgh \] Where: - \( m = 0.005 \, \text{kg} \) (mass) - \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity) - \( h \) is the height we want to find. Setting the potential energy equal to the gravitational potential energy: \[ 0.1 \, \text{J} = 0.005 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times h \] ### Step 4: Solve for Height (h) Rearranging the equation to solve for \( h \): \[ h = \frac{0.1 \, \text{J}}{0.005 \, \text{kg} \times 9.81 \, \text{m/s}^2} \] Calculating the denominator: \[ 0.005 \times 9.81 = 0.04905 \, \text{kg m/s}^2 \] Now substituting back into the equation for \( h \): \[ h = \frac{0.1}{0.04905} \approx 2.04 \, \text{m} \] ### Conclusion The height to which the mass is projected is approximately 2.04 m. Since the closest option provided is: **C. 2 m** ### Explanation of Other Options - **A. 8 m**: This option is too high. The energy calculations show that the mass cannot reach this height given the energy stored in the spring. - **B. 4 m**: This option is also too high for the same reason as option A. The energy from the spring does not support this height. - **D. 1 m**: This option is too low. The calculations indicate that the mass will rise higher than this value. ### Revision Summary - The potential energy stored in a spring is calculated using \( PE = \frac{1}{2} k x^2 \). - When released, this energy converts to gravitational potential energy \( GPE = mgh \). - The height can be found by equating the two energies and solving for \( h \). - The correct answer is approximately 2 m, which corresponds to option C.