A sphere of mass m = 2.5 kg sits on a horizontal, frictionless surface. Attached

Question

A sphere of mass m = 2.5 kg sits on a horizontal, frictionless surface. Attached to the sphere is an ideal spring with spring constant k = 33 N/m. At time t = 0 the mass is pulled aside from the equilibrium position a distance d = 11 cm (in the positive direction) and released from rest. After this time, the system oscillates between x = plusminus d. (a) Determine the magnitude of the force (in Newtons) required to initially displace the mass d = 11 cm from equilibrium. (b) What is the sphere's distance from equilibrium, in meters, at time t = 1 s in m? x (1) =| (c) Determine the frequency (in Hertz) with which the spring-mass system will oscillate after released. (d) Calculate the maximum speed v_max in m/s obtained by the sphere. (e) At what point(s) in the motion does the sphere reach maximum speed? Select all that apply. (f) Calculate the magnitude of the maximum acceleration a_max in m/s^2 experienced by the sphere. (g) At what point(s) in the motion does the sphere experience maximum acceleration? Select all that apply.

Explanation / Answer

mass, m=2.5 kg

spring constant, K=33 N/m

amplitude, A=d=11cm =11*10^-2 m

a)

F=k*d

=33*0.11

=3.63 N

b)

x=A*sin(w*t)

here,

W=sqrt(K/m)

W=sqrt(33/2.5)

W=3.63 rad/sec

now,

x=0.11*sin(3.63*1)

x=0.069 m or 6.9cm

c)

frequemcy, f=2pi/w

f=2pi/3.63

f=1.73 Hz

d)

Vmax=A*W

=0.11*3.63

=0.399 m/sec or 0.4 m/sec

e)

at x=0 ( mean positon),

speed v becomes maximum

f)

a_max=W^2*A

=3.63^2*0.11

=1.45 m/sec^2

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