Simple Harmonic Motion
It is the periodic (repeated) motion. in which a particle moves to and fro motion about a mean (equilibrium) position under a restoring force(acceleration) which is always directed toward mean position such that its magnitude is proportional to displacement (a∝ -y).
Or, F ∝ -y
F= -ky
Types: (i) Linear SHM; a∝-y
Or, a=-ky
Or,
(ii) Angular SHM
∝ = -Kθ
Equation of Simple Harmonic Motion in different conditions
The general equation of SHM is α= Asin(wt+).
1) From position (x=0, t=0) to x=+A
X= Asinwt ( =0)
2) From right extreme (t=0, x=+A)
x= A sin (wt + π/2)
x= A coswt
3) From left extreme (t=0, x=-A)
x= A sin(wt - \2)
x= -Acoswt
4) from half of maximum displacement to A
x= Asin(wt +/6) as,
= sin-1(x/a) where x = A/2
Time taken by the particle
1) from (x=0 to A/2)
x = A sinwt
or, A/2=Asinwt
t=T/12
2)from (x=A to A/2)
x = A cosw t
Or,A/2=Acoswt
t=T/6
3)from one extreme to another extreme.(x=-Ato A)
t=T/2
4)From mean position to extreme position,
x = A sinwt
T= T/4
Where, T isthe total time period from A to-A and from -A to A.
Velocity in SHM
V=
So, at x=0
So,
and at x = A
Also, v2=w² (A2-x²)
Or,
Or,
which shows graph between velocity and displacement on SHM is elliptical.
Acceleration on SHM
•
So, graph between a and x is st. line.
•At x=0, a=0 (min), at v=A; a=-w²A.
• For maximum amplitude a0 and maximum velocity V0 Amplitude and angular velocity are given as so,
Spring mass system
•for both horizontal and vertical arrangements
F=-kx=ma
Or,
•If mass of sparing is considered ();
Reduced Mass
It is the behaviour of combined two masses when they are attracted toward each other by a certain force.
**** Distance doesn't matters.
For two bodies joined as aside:
Time period is where μ=reduced mass=
Spring combination
Simple Pendulum
The time period of pendulum is given by;
for a very short pendulum; R=Radius of earth ;l<<< so, For ; For L=R, =59.8min. Note time period is independent of mass of bob of pendulum. Change on length •If the length of pendulum increases, time period increases as (Tα); motion becomes slow and time with loose and vice versa. (i)If initial time period is T1, and by some factor it changes to T₂ if (T2>T1)•Loss in time =*86400 secs. (ii)if (T2>T1)gain in time =*86400 secs. •If a simple pendulum is taken from mean sea level to height h, Loss in time = 13.5*h secs.(h in km) •If a simple pendulum is taken from mean sea level to depth h, Loss in time =13.5*hsecs. (h in km) •Due to change on temperature loss or gain=0.5αΔθ Change in acceleration due to gravity where is the effective acceleration due to gravity. In a lift i. Moving upward: ii. Moving downward: iii. Moving with constant velocity: On inclined plane where is angle made by inclined plane with horizontal In a train i. Accelerating with acceleration : ii. Moving with constant speed in circular path: Tension on string we have, so = = where A is the amplitude Energy in simple harmonic motion P.E.==== K.E.=== T.E.== constant So,
P.E α α α == K.E α (A2-x2) α = == Hence, K.E and P.E has double the frequency of SHM. Examples solution: F=mrw2 = α f2 so,and , for,
(say) or, ,f is required frequency. Solution: Initial Energy= Final energy so, Taking +ve as > mg