1. The motion of a body is described by the equation ay = −308y.
(a) Find the angular frequency.
(b) Find the frequency.
(c) Find the period.
2. A 2.10 kg mass attached to a spring oscillates with a period of 0.395 s and an amplitude of 20.5 cm.
(a) Find the total mechanical energy of the system
(b) Find the maximum speed of the mass.
3. The motion of a body is described by the equation
1.90 sin (0.180πt)
where t is in s and y is in m.
(a) Find the amplitude.
(b) Find the period.
(c) Find the frequency.
(d) Find the displacement at t = 5.00 s.
(e) Find the displacement at t = 26.0 s.
To solve these problems, we will use the formulas and concepts from simple harmonic motion.
1. The equation describing the motion of the body is given as ay = -308y.
(a) The angular frequency (ω) is the coefficient of y in the equation, so ω = 308 rad/s.
(b) The frequency (f) is related to the angular frequency by the equation f = ω/(2π), where 2π is approximately 6.28. Therefore, f = 308/6.28 ≈ 49 Hz.
(c) The period (T) is the inverse of the frequency, so T = 1/f ≈ 1/49 ≈ 0.020 s.
2. The period (T) of the oscillation is given as 0.395 s and the amplitude (A) is given as 20.5 cm.
(a) The total mechanical energy of the system (E) is given by E = (1/2)mv^2 + (1/2)kA^2, where m is the mass, v is the velocity, k is the spring constant, and A is the amplitude. Since the velocity at the maximum displacement is zero, the total mechanical energy is equal to (1/2)kA^2. Therefore, we need to determine the spring constant to find the total mechanical energy.
(b) The maximum speed of the mass can be found using the formula v = ωA, where ω is the angular frequency and A is the amplitude. We need to determine the angular frequency to find the maximum speed.
3. The motion of the body is described by the equation 1.90 sin(0.180πt), where t is in seconds and y is in meters.
(a) The amplitude (A) of the motion is the coefficient of the sin function, so A = 1.90 m.
(b) The period (T) is the time taken for one complete oscillation, which can be found using the formula T = 2π/ω, where ω is the angular frequency. We need to determine the angular frequency to find the period.
(c) The frequency (f) is the inverse of the period, so f = 1/T. We need to determine the period to find the frequency.
(d) The displacement at t = 5.00 s can be found by substituting the value of t into the equation and evaluating it: y(5.00) = 1.90 sin(0.180π * 5.00).
(e) The displacement at t = 26.0 s can be found using the same process as in part (d): y(26.0) = 1.90 sin(0.180π * 26.0).