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Oscillations Examples I

1. When we put on spring a body with weight of 20 N and spring stretches for 40 cm. If we put on same spring the body with weight of 10N the spring stretches for 35 cm. Calculate the spring elasticity constant? What is the length of upstretched spring? 
G1=20 Ny1+l0=40 cm=0.4 mG2=10 Ny2+l0=35 cm=0.35 m_l0=?,k=?y1=0.4l0y2=0.35l0ky1=G1ky2=G2G1y1=G2y2G1y2=G2y120(0.35l0)=10(0.4l0)720l0=410l010l0=3l0=0.3 m=30 cmy1=0.40.3=0.1 m=10 cmy2=0.350.3=0.05 m=5 cmky1=G1k=G1y1=200.1=200 N/m

2.The body hung on a spring vibrates harmonically with circular frequency of 0.5 rad/s. Amplitude of harmonic vibrations is 0.8 m. Derive the equation which describes the relation between elongation y and time t, speed u and time t and acceleration a and time t if at the moment t = 0 body moves from equilibrium position in positive direction of y-axis. 
ω=0.5 rad/sA=0.8 m _y=?,v=?,a=?y=Asin(ωt+θ)θ=0y=0.8sin(0.5t)v=dydt=Aωcos(ωt)v=0.80.5cos(0.5t)v=0.4cos(0.5t)a=dvdt=d2ydt2=Aω2sin(ωt)a=0.80.52sin(0.5t)a=0.2sin(0.5t)


3. If the body vibrates harmonically with amplitude of 4 cm determine the distance that body travels during one period. 
A=4 cm=0.04 mt=T _s=4A=40.04=0.16 ms=16 cm

4. Equation of the body with harmonic oscillations is given by: 
y=0.02sin(π2t+π4)
a)Amplitude.From given equation the amplitude is: 
A=0.02 m=2 cm
b)Circular frequency and period. From given equation the circular frequency is: 
ω=π2T=2πω=2ππ2=4s
  c) Maximum speed. To find the maximum speed first we need to derive the equation of speed and then plot it in order to visualize
y=0.02sin(π2t+π4)u=dydt=0.02π2cos(π2t+π4)u=0.0314159cos(π2t+π4)
The u,t graph is shown in the next figure. 

u=0.0314159cos(π2t+π4)0.0314159cos(π2t+π4)=0.0314159/:0.0314159cos(π2t+π4)=1/arccosπ2t+π4=2πn/42πt+π=8πn2πt=8πnπfor n=12πt=7πt=72=3.5sNow let  s get back into the equation of speed and substitute time tu=0.0314159cos(π272+π4)u=0.0314159cos(7π4+π4)u=0.0314159cos(8π4)=0.0314159cos(2π)=0.0314159 m/su=3,14159 cm/s
d)Maximum acceleration. In order to obtain the maximum acceleration first we need to obtain the expression for acceleration in other words we need to derivate expression for speed 
u=0.0314159cos(π2t+π4)a=dudt=0.0314159π2sin(π2t+π4)a=0.04934798sin(π2t+π4)t=?0.04934798sin(π2t+π4)=0.04934798/(0.04934798)sin(π2t+π4)=1/arcsinπ2t+π4=π2+2πnIf n=0π2t+π4=π2/42πt+π=2π2πt=π/2πt=12=0.5sa=0.04934798sin(π2t+π4)=0.04934798m/sa=4.9347cm/s
e)Initial phase angle 
θ0=π4
5. Elongation of the body which harmonically oscillates in time is shown in next figure. Determine the oscillation amplitude, period, circular frequency and phase angle. Write down the equation of harmonic oscillation motion, speed and acceleration.



A=4mT=4s_ω=2πT=2π4=π2rad/sθ0=πy=Asin(ωt+θ0)y=4sin(π2t+π)u=Aωcos(ωt+θ0)u=4π2cos(π2t+π)u=2πcos(π2t+π)a=Aω2sin(π2t+π)a=4π24sin(π2t+π)a=π2sin(π2t+π)


6. Figure shows body with mass m attached on the spring. The spring is stretched for 5 cm and then we let her go and the body oscillates with period of 2 s.

Determine:
a)The equation of oscillating body if at the time t= 0 body was in position B and was moving towards the position A.
b)The equation of oscillating body if at the moment t= 0 body was at position A.
c)The equation of oscillating body if at the moment t= 0 body was at position C.
d)Highest speed is at position?
e)Highest acceleration is at position?
f)Kinetic energy of the body is highest at position?
g)Potential energy is highest at position?
Solution
A=5cm=0.05mT=2sω=2πT=πrad/s_a)y=f(t)=? t=0s at point B moving to point Ab)y=f(t)=? t=0s at point Ac)y=f(t)=? t=0s at point Cd)Maximum speede)Maximum accelerationf)Kinetic energy maximum?g)Potential energy?
a) The equation of oscillating body if at the time t= 0 body was in position B and was moving towards the position A.
y=Asin(ωt+θ0)y=0.05sin(πt)t=0sy=0m
b)The equation of oscillating body if at the moment t= 0 body was at position A. 
y=Asin(ωt+θ0)y=0.05sin(πt+π2)t=0sy=0.05m
c)The equation of oscillating body if at the moment t = 0 body was at position C.
y=Asin(ωt+θ0)y=0.05sin(πt+3π2)t=0sy=0.05m
d)Highest speed is at position?
y=0.05sin(πt)u=dydt=0.05πcos(πt)u=0,157cos(πt)0,157cos(πt)=0,157cos(πt)=1πt=2πnn=0t=0su=0,157cos(0)=0,157m/s
Maximum speed is at positon A, and C.
e)Maximum acceleration 
u=dydt=0.05πcos(πt)a=0.05π2sin(πt)a=0.49sin(πt)

a=0.49sin(πt)0.49sin(πt)=0.49sin(πt)=1πt=3π2t=32=1.5sa=0.49sin(32π)=0.49m/s2
f)Maximum kinetic energy of the system 
Ek=mω22y20cos2(ωt+θ)Assumption m=1kgEk=0.012337J

The maximum kinetic energy is at point B. 
g)Potential energy is highest at position?
Ep=mω22y20sin2(ωt+θ)Assumption m=1kgEp=0.00125J
Maximum potential energy is highest at position A and C. 


7.Body of mass 0.4 kg is attached on spring and oscillates around equilibrium position with frequency 2 Hz and amplitude 20 cm.
a)      Derive the equation that describes the oscillation of the body if at the moment t= 0 the body was at equilibrium position and was moving in positive direction of y-axis.
b)      Derive the equation that describes the motion of the body if at the moment t = 0 the body was 20 cm from equilibrium position in negative direction of y-axis.
c)      Determine the speed of the moving body when it goes through the equilibrium position
d)      Derive the equation for body oscillation speed in dependence of time for cases a and b and draw v,t graphs
e)      Determine the speed of the body when the body is 10 cm from equilibrium position.
f)       Derive the equation for acceleration of the moving body for cases a) and b)  and draw a,t graphs
g)      Determine the total energy of the system
h)      Determine the maximum force that is acting on the body
i)        In which position body has maximum speed and in which maximum acceleration. What is the ratio between acceleration and the speed in each moment? Does this ratio changes over time ?
j)        What is the ratio between acceleration and elongation in each moment?
k)      Draw a graph which shows how speed changes over elongation u =f(y).
Solution:
m=0.4kgA=20cm=0.2mf=2HzT=1/f=1/2sω=2πT=4πrad/s_a)y=f(t)=? t=0s yt-graphb)y=f(t)=? t=0s at point A!!!! ovo sve treba izmjenitic)y=f(t)=? t=0s at point Cd)Maximum speede)Maximum accelerationf)Kinetic energy maximum?g)Potential energy?

a)Derive the equation that describes the oscillation of the body if at the moment t= 0 the body was at equilibrium position and was moving in positive direction of y-axis.
y=Asin(ωt+θ0)y=0.2sin(4πt)


b)Derive the equation that describes the motion of the body if at the moment t = 0 the body was 20cm from equilibrium position in negative direction of y-axis.
y=Asin(ωt+θ0)y=0.2sin(4πtπ2)




c) Determine the speed of the moving body when it goes through the equilibrium position

 y=0.2sin(4πt)u=dydt=0.24πcos(4πt)u=2.51cos(4πt)t=0su=2.51m/s

d)Derive the equation for body oscillation speed in dependence of time for cases a and b and draw u,t graphs For the case a the speed of body oscillation is: 
 u=2.51cos(4πt)
For the case b the equation which describes the body oscillation speed can be written in the following form: 
y=0.2sin(4πtπ2)u=dydt=2.51cos(4πtπ2) 

e)Determine the speed of the body when the body is 10 cm from equilibrium position.

 y=0.2sin(4πt)0.2sin(4πt)=0.1sin(4πt)=0.54πt=π6t=124=0.0416667su=2.51cos(4πt)u=2.51cos(π6)=2.173m/s

 f)Derive the equation for acceleration of the moving body for cases a) and b) and draw a,t graphs For the case a the equation of acceleration:
 u=2.51cos(4πt)a=dudt=2.514πsin(4πt)a=31.5159sin(4πt) 

 For the case b the equation of acceleration: 

 u=2.51cos(4πtπ2)a=dudt=2.514πsin(4πtπ2)a=31.5159sin(4πtπ2)


8.The amplitude of harmonic oscillation is 30cm and period 2s. In moment t= 0s elongation is 0 and the body moves in positive direction of y-axis. Determine the elongation, speed and acceleration at the moment t=T/12. 
A=30cm=0.3mT=2st=T/12=1/6s_y=?v=?a=?y=Asin(ωt)ω=2πT=2π2=π rad/s y=0.3sin(πt)=0.3sin(π6)=0.15 my=15 cmv=Aωcos(ωt)=0.942478cos(πt)v=0.942478cos(π6)=0.8162 m/sv=81.62 cm/sa=2.96088sin(ωt)=2.96088sin(π6)a=1.48044 m/s =148.044 cm/sa=|148.044|=148.044 cm/s
9.The body is attached to a spring and is moved from equilibrium position for 10cm than is left to oscillate. At what distance from the equilibrium position will:
a) Speed of the body be equal to one half of maximum speed?
b)Acceleration of the body be equal to one half of maximum acceleration? 
A=10cm=0.1m_a)v=0.5vmaxy=?y=Asin(ωt)v=Aωcos(ωt)v=vmaxcos(ωt)=1ωt=α=2nπn=0α=0v=0.5vmaxcos(ωt)=0.5ωt=α=π3y=Asin(ωt)=0.1sin(60)=0.0866 my=8.66 cmb)a=0.5amaxa=amaxsin(ωt)=1ωt=α=nπ2n=1α=π2a=0.5amaxsin(ωt)=0.5ωt=α=π6y=Asin(ωt)=0.10.5=0.05 my=5 cm 10.The body harmonically oscillates with the circular frequency of 0.5 rad/s. Amplitude of the oscillations is 80 cm.
a)Derive the equations of elongation, speed and acceleration if at the moment t=0 the body was in equilibrium position and was moving in positive direction of y-axis.
b)Derive the equation that connects speed and elongation. 
ω=0.5rad/sA=80cm _a)y=Asin(ωt)y=0.8sin(0.5t)u=dydt=Aωcos(ωt)u=0.4cos(0.5t)a=dudt=Aω2sin(ωt)a=0.2sin(0.5t)b)y=Asin(ωt)sin(ωt)=yAv=Aωcos(ωt)cos(ωt)=vAωsin2(ωt)+cos2(ωt)=1y2A2+v2A2ω2=1/A2ω2y2ω2+v2=A2ω2v2=ω2(A2y2)v1,2=± ω A2y2v1,2=± 0.5 0.64y2



11.The period of harmonic oscillations is 3.6 s determine the shortest time what is needed for the body which oscillates to distance itself from equilibrium position for half amplitude. 
T=3.6 sy=0.5A_t=?ω=2πf=2πT=2π3.6=1.74 rad/sy=Asin(ωt)0.5A=Asin(1.74t)0.5=sin(1.74t)1.74t=arcsin(0.5)1.74t=π6t=π61.74=0.3 s

12.Body of mass 1 kg harmonically oscillates according to the equation y = 0.32 sin(7.4 t). Determine the:
a) amplitude, 
b) frequency and 
c) kinetic and potential energy of the body when the body is 0.26 m from equilibrium position. 
m=1 kgy=0.32sin(7.4t)_a)A=?A=0.32 mb)f=?ω=2πff=ω2π=7.42π=1.18 Hzc)y=0.26 mEk=?Ep=?y=0.32sin(7.4t)0.260.32=sin(7.4t)7.4t=arcsin(0.8125)t=54.347.4=7.34sEk=mω22A2cos2(ωt+θ)Ek=17.4220.322cos2(7.47.34)=1.059 JEp=mω22A2sin2(ωt+θ)=17.4220.322sin2(7.47.34)Ep==1.74468 J

13.What variables in harmonic oscillations in amplitude position have the maximum value?
Elastic energy of a spring in dependence on time is shown in next figure. 

Elastic energy in dependence on time is shown in following figure. 
From previous figures we see that the system at amplitude position has the maximum energy. 
The following figures show elongation, speed and acceleration in dependence of time. 

As we can see from previous figure the maximum value at amplitude positions is acceleration. 
The dependence of potential and kinetic energy of time is show in following figure. 

From previous figures the maximum values at amplitude position are Elastic force, Potential Energy and Acceleration.

14. Body of mass 1 kg is attached to the horizontally laid spring with spring constant of 120 N/m. At the moment t = 0 the body is moved so spring is compressed. The initial speed of the body is 3 m/s. If friction between body and a surface is neglected determine:
a)Period and frequency of oscillations
b)Amplitude
c)Maximum acceleration
d)Total energy
m=1 kgk=120N/mt=0 sv=3m/s_a)T=?f=?T=2πmk=2π1120=0.57 sf=1T=10.57=1.74346 sb)A=?ω=2πf=10.9545 rad/sv=Aωcos(ωt)A=vωcos(ωt)=310.9545cos(10.95450)A=310.9545=0.27386 mc)amax=?a=amaxy=At=T4,3T4t=0.143393 sa=Aω2sin(ωt)a=0.2738610.95452sin(10.95450.143393180π)a=amax=32.86 m/s2d)E=mω22A2=110.9545220.273862=4.5 J

15.The body of mass 0.4 kg is attached on spring and oscillates with frequency of 3 Hz. Determine the frequency of oscillations if the mass of the body is 0.1 kg?
m1=0.4 kgf1=3 Hzm2=0.1 kg_f2=?k1=k2=kT1=1f1T1=2πm1k1f1=2πm1kk=4f21m1π2k=4320.4π2=142.122 N/mf2=12πm2k=12π0.1142.122=6 Hz

16.Elastic spring is transferred to a planet where gravitational acceleration is equal to one half of gravitational acceleration on earth. The period of spring oscillation when compared to period oscillation on Earth will be:
a)Unchanged
b)Two times bigger
c)Two times smaller
d)Four times bigger
e)Four times smaller
gp=0.5gearth_TearthTP=?T=2πmk=2πGgkTearthTP=GearthgearthGpgp=GearthgearthGp0.5gearth=0.5GearthGpmearth=mpGearthgearth=GpgpGearthgearth=Gp0.5gearthGp=0.5GearthTearthTP=0.5GearthGp=0.5Gearth0.5Gearth=1Tearth=Tp
Answer: the periods of oscillations are the same or unchanged.
17.Spring has attached weight and the system oscillates and makes 45 oscillations per minute. What to do with the mass of a weight so the system would oscillate with 30 oscillations per minute ?

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