A 10kg particle undergoes simple harmonic motion with an amplitude of 2.0mm, a maximum acceleration of 8.0x10^3 m/s^2, and an unknown phase constant (phi) What are:
a.) the period of the motion
b.) the maximum speed of the particle
c.) total mechanical energy of the oscillator
What is the magnitude of the force on the particle when the particle is at:
d.) its maximum displacement
e.) half its maximum displacement
To find the answers to these questions, we can use the formulas and equations related to simple harmonic motion.
a.) The period of the motion can be found using the formula: T = 2π/ω, where T is the period and ω is the angular frequency. In simple harmonic motion, the angular frequency is given by the formula ω = √(k/m), where k is the spring constant and m is the mass of the particle. In this case, we are given the mass of the particle (10kg), but we need to find the spring constant. The maximum acceleration (a_max) is related to the angular frequency by the formula a_max = ω^2A, where A is the amplitude of the motion. Rearranging this equation gives ω = √(a_max/A). Thus, the angular frequency is ω = √(8.0x10^3 m/s^2 / 0.002m). Substituting this value into the period formula, we find T = 2π / (√(8.0x10^3 m/s^2 / 0.002m)). Evaluting this expression, we get the period T.
b.) The maximum speed of the particle can be found using the formula vmax = ωA, where vmax is the maximum speed of the particle, ω is the angular frequency, and A is the amplitude. We have already found ω in the previous step, and the amplitude (A) is given as 2.0mm. Substituting these values into the formula gives vmax = (√(8.0x10^3 m/s^2 / 0.002m)) * 0.002m. Simplifying this expression gives the maximum speed vmax.
c.) The total mechanical energy of the oscillator is given by the formula E = (1/2)kA^2, where E is the total mechanical energy, k is the spring constant, and A is the amplitude. We need to find the spring constant (k) in order to calculate the mechanical energy. The spring constant can be found using the equation k = mω^2, where m is the mass of the particle and ω is the angular frequency. We have already found ω in the first step, and the mass (m) is given as 10kg. Substituting these values into the formula gives k = 10kg * (√(8.0x10^3 m/s^2 / 0.002m))^2. Using this value of k, we can then calculate the mechanical energy E.
d.) When the particle is at its maximum displacement (amplitude), the force acting on it can be found using Hooke's law: F = -kx, where F is the force, k is the spring constant, and x is the displacement from equilibrium. Given that the displacement is the amplitude (2.0mm), and we have already found the spring constant (k), we can calculate the force using the formula F = -k * 2.0mm.
e.) When the particle is at half its maximum displacement, the force acting on it can be found using the same formula as above, F = -kx. In this case, the displacement is half of the amplitude (1.0mm). We can now calculate the force using the formula F = -k * 1.0mm.
By following these steps, you should be able to find the answers to each part of the question.