A240 g block connected to a light spring for which the force constant is 9.50 n/m is free to oscillate on a horizontal, frictionless surface. the block is displaced 5.00cm from equilibrium and released from rest as in the figure.

(a) find the period of its motion.

(b) determine the maximum speed of the block.

(c) what is the maximum acceleration of the block?

(d) express the position, velocity, and acceleration as functions of time.

solve it

(a) find the period of its motion.

conceptualize study the figure and imagine the block moving back and forth in simple harmonic motion once it is released. set up an experimental model in the vertical direction by hanging a heavy object such as a stapler from a strong rubber band.

categorize the block is modeled as a particle in simple harmonic motion. we find values from equations developed in this section for the particle in simple harmonic motion model, so we categorize this example as a substitution problem.

use the equation to find the angular frequency of the block-spring system:

ω =
sqrt3a. gif
k/m
=
sqrt4a. gif
(9.50 n/m)/240*10-3
= 6.292 rad/s
use the equation to find the period of the system:

t =
2π/ω
=
2π/6.292rad/s = .99985
(b) determine the maximum speed of the block.

use the equation to find vmax:

vmax = ωa = (6.292 rad/s)(5.00 multiply. gif 10-2 m) = .3146 m/s
(c) what is the maximum acceleration of the block?

use the equation to find amax:

amax = ω2a = (6.292 rad/s)2(5.00 multiply. gif 10-2 m) = 1.9794 m/s2
(d) express the position, velocity, and acceleration as functions of time.

find the phase constant from the initial condition that x = a at t = 0:

x(0) = a cos ϕ = a → ϕ = 0

use the equation and the values of ω and ato write an expression for x(t):

x = a cos(ωt + ϕ)

= .05·cos(6.292t)

use the equation and the values of ω and ato write an expression for v(t):

v = −ωa sin(ωt + ϕ)

=

use the equation and the values of ω and ato write an expression for a(t):

a = −ω2a cos(ωt + ϕ)

= −1.98cos(6.292t)

master ithints: getting started | i'm stuck!

use the equations below. (note that the direction is indicated by the sign in front of the equations.)

x = (0.0526 m) cos(6.292t + 0.10π)
v = −(0.331 m/s) sin(6.292t + 0.10π)
a = −(2.082 m/s2) cos(6.292t + 0.10π)
(a) determine the first time (t > 0) that the position is at its maximum (positive) value.


(b) determine the first time (t > 0) that the velocity is at its maximum (positive) value.