A system consisting of a small 1.17-kg object attached to a light spring oscilla

Question

A system consisting of a small 1.17-kg object attached to a light spring oscillates on a smooth, horizontal surface. A graph of the position x of the object as a function of time is shown in the figure below. Use the graph to answer the following questions.

(a) What are the angular velocity 0 and the frequency of the motion?

(b) What is the spring constant of the spring?
kg/s2

(c) What is the maximum speed of the object?
cm/s

(d) What is the maximum acceleration of the object?
m/s2

Explanation / Answer

mass of the object, m=1.17 kg

assuming time t=0.28 sec

from the graph, time period, T=2*t

T=2*(0.28) = 0.56 sec

part a:   from definiton of Anguar velocity W=2pi f or 2pi/T

so W = 2pi/T = 2pi/0.56

W=11.22 rad/sec

as W = 2pif

f = W/2pi = 11.22/2pi = 1.786 Hz

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part 2)

formula for time perion T = 2pi*Sqrt(m/K)

m is mass = 1.17 kg

K is Spring Constant = ?

T^2 = 4pi^2 * m/k

k = 4pi^2* m/T

K = 4*pi^2 * 1.17/0.56

K = spring constant = 82.48 N/m

------------------

part C:

for max speed , use   maximum speed Vmax=A*w

where A is amplitude, A=4cm

Vmax=0.04*11.22

Vmax=0.45 m/s or 45 cm/s

----------------------------

part d:

for maximum acceleration use the formula a=w^2*A

a=11.22^2*(0.04)

a=5.03 m/s^2
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