Final answer:
To find the mass of a ball oscillating on a spring with a known spring constant and measured displacement and speed, we use the conservation of energy in simple harmonic motion. Calculating the potential and kinetic energies at the point of interest and equating them to the energy at the maximum displacement, the ball's mass is found to be 0.0296 kg, or 29.6 grams.
Step-by-step explanation:
The student asked about the mass of a ball attached to a spring with a known spring constant and a specific displacement and speed at a given point. To find the mass, we can use the principles of conservation of energy in simple harmonic motion. The total mechanical energy (sum of potential and kinetic energy) is constant at any point during the motion.
Firstly, the potential energy (PE) stored in the spring when the ball is 12 cm (0.12 m) away from the equilibrium position is given by PE = 0.5 × k × x^2, where k is the spring constant and x is the displacement. The kinetic energy (KE) at the same point is given by KE = 0.5 × m × v^2, where m is the mass we need to find and v is the velocity. At maximum displacement, the kinetic energy is zero and all the energy is stored as potential energy.
Using the conservation of energy, we have:
PE_{max} = PE + KE
0.5 × k × A^2 = 0.5 × k × x^2 + 0.5 × m × v^2
We can solve for mass m:
m = (k × (A^2 - x^2)) / v^2
m = (25 N/m × (0.18^2 - 0.12^2) m^2) / (3.9 m/s)^2
m = (25 × (0.0324 - 0.0144)) / 15.21
m = (25 × 0.018) / 15.21
m = 0.45 / 15.21
m = 0.0296 kg
The mass of the ball is 0.0296 kg or 29.6 grams.