Question

A particle with a mass of 0.230 kg is attached to a horizontal spring with a spring constant of 8.28 N/m. At the moment t = 0, the particle has its maximum speed of 9 m/s and is moving to the left. (Assume that the positive direction is to the right.)

(a) Write an expression for the particle's motion as a function of time t. Express all numerical quantities in terms of SI units (including radians for angles), but do not include units in the expression.
(b)Where in the motion is the potential energy of the spring-particle system three times the kinetic energy? (Note there are two positions, one positive and one negative—enter only the positive value. Enter your answer in m.)
(c)What is the minimum time interval (in s) required for the particle to move from x = 0 to x = 1.00 m? (Be sure to enter the minimum time and not the total time elapsed from t = 0.)
(d)Calculate the length of a simple pendulum (in m) with the same period as the spring-particle system.

Answer 1

The equation for the particle's motion as a function of time t is x(t) = A*cos(ωt + φ). The potential energy of the spring-particle system is three times the kinetic energy when the particle is at the extreme positions of its motion. The minimum time interval required for the particle to move from x = 0 to x = 1.00 m can be calculated using the equation t = (1/ω)*acos((x - A)/A).

Step-by-step explanation:

(a) The equation for the particle's motion as a function of time t can be determined using the equation for simple harmonic motion:

x(t) = A*cos(ωt + φ)

where x(t) is the displacement of the particle from its equilibrium position, A is the amplitude of the motion, ω is the angular frequency given by ω = √(k/m), t is the time, and φ is the phase angle.

(b) The potential energy of the spring-particle system is three times the kinetic energy when the particle is at the extreme positions of its motion, where the displacement from the equilibrium position is equal to the amplitude of the motion, A. Therefore, the potential energy is three times the kinetic energy when x = A.

(c) The minimum time interval required for the particle to move from x = 0 to x = 1.00 m can be calculated using the equation:

t = (1/ω)*acos((x - A)/A)

(d) The length of a simple pendulum with the same period as the spring-particle system can be determined using the equation:

L = g*T^2/(4*pi^2)