Part A: A mass is attached to the end of a spring and set into simple harmonic m
ID: 2302843 • Letter: P
Question
Part A:
A mass is attached to the end of a spring and set into simple harmonic motion with an amplitude A on a horizontal frictionless surface. Determine the following in terms of only the variable A.
(a) Magnitude of the position (in terms of A) of the oscillating mass when its speed is 60% of its maximum value.
(b) Magnitude of the position (in terms of A) of the oscillating mass when the elastic potential energy of the spring is 60% of the total energy of the oscillating system.
Part B: A mass of 417 g is attached to a spring and set into simple harmonic motion with a period of 0.256 s. If the total energy of the oscillating system is 7.14 J, determine the following.
(a) maximum speed of the object
m/s
(b) force constant
N/m
(c) amplitude of the motion
m
Explanation / Answer
amplitude=A
a)
we know the relation for velocity v=w*sqrt(A^2-x^2)
maximum speed v=wA
60% of maximum value
0.6wA=w*sqrt(A^2-x^2)
0.6A =sqrt(A^2-x^2)
0.36A^2=A^2-x^2
x^2 =0.64 A^2
x =0.8 A
b)
potential energy =0.5kx^2
total energy =0.5kA^2
60% of total E =0.6*0.5*kA^2 =0.3kA^2
0.3KA^2=0.5kx^2
x =0.774 A
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partB:
mass m=0.417 kg
T=0.256 s
total energy E =7.14 J
a)
E=0.5mv^2
7.14=0.5*0.417v^2
v =5.85 m/s
b)
w=2pi/T =6.28/0.256 = 24.53 rad/s
force constant k=mw^2 =0.417*24.53^2 =250.9 N/m
c)
w=2pi/T =6.28/0.256 = 24.53 rad/s
E=0.5*mW^2A^2=7.14
0.5*0.417*24.53^2*A^2 =7.14
A =0.238 m =23.8 cm