Question

A particle with a mass of 0.300 kg is attached to a horizontal spring with a spring constant of 43.2 N/m. At the moment t = 0, the particle has its maximum speed of 30 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. x = 2.5 cos 12t + (( 5 )) m (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.) X = + 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.) S (d) Calculate the length of a simple pendulum (in m) with the same period as the spring-particle system. m

Answer 1

The expression for the particle's motion is x = 2.5 cos(12t) + 5 m. The potential energy is three times the kinetic energy when x = 2.5 m. The minimum time interval for the particle to move from x = 0 to x = 1.00 m is approximately 0.174 s. The length of a simple pendulum with the same period as the spring-particle system is approximately 2.52 m.

Step-by-step explanation:

(a) The expression for the particle's motion as a function of time t is x = 2.5 cos(12t) + 5 m. This represents the equation of motion for a particle attached to a horizontal spring. The particle's position is given by the cosine function, with an amplitude of 2.5 meters, a frequency of 12 Hz, and a phase shift of 0 radians.

(b) The potential energy of the spring-particle system is three times the kinetic energy when the particle is at its maximum displacement from equilibrium. This occurs when x = 2.5 meters.

(c) To calculate the minimum time interval required for the particle to move from x = 0 to x = 1.00 m, we can set up the equation x = 2.5 cos(12t) + 5 = 1.00 m and solve for t. By rearranging the equation and solving for t, we find that the minimum time interval is approximately 0.174 seconds.

(d) The period of a simple pendulum is given by the equation T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. To find the length of the pendulum with the same period as the spring-particle system, we can rearrange the equation to solve for L. Using the period of the spring-particle system (T = 1/f = 1/12 s) and the value of g, the length of the pendulum is approximately 2.52 meters.

Learn more about Expressing Particle's Motion in a Spring System