Question

Where does the equation for simple pendulum motion come from? \[T=2\pi \sqrt{\frac{ l }{ g }}\] where T is period, l is string length, g is acc. due to gravity

Answer 1

write down the equation of motion for pendulum

Answer 2

why is the mg force not straight down?

Answer 3

sorry...mg is straight down only

Answer 4

does it help u in finding the time period?

Answer 5

I want to know how the formula is derived...

Answer 6

I know
write down the equation of motion along the tangent...
m( a_t) = mg sin(theta)
a_t---tangential acc

Answer 7

it's fine?

Answer 8

net force along tangent is mg sin (theta)

Answer 9

yes

Answer 10

now v_tangential= r ( angular vel)

Answer 11

differentiate it
a_tangential= r(angular acc)

Answer 12

ok

Answer 13

r is the length of pendulum in this case

Answer 14

now substitute a_tangential into
m( a_t) = mg sin(theta)

Answer 15

m(r*angular_acc) = m(sin(theta))

Answer 16

m ( l(angular acc))= mg sin(theta)....l=r= length of pendulum

Answer 17

now l(angular acc))= g sin(theta)

Answer 18

where did the l come from?

Answer 19

l=r=length of pendulum...pendulum is like bob is moving in a circle pivoted at centre

Answer 20

oh right... l = r. I forgot about that

Answer 21

now acceleration = g/l sin(theta)

Answer 22

yep

Answer 23

equation of SHM is
a= - w^2 x
then time period T= 2 pi/w

Answer 24

I havent learnt SHM yet.
whats w?

Answer 25

w(omega)---angular frequency
acceleration = g/l sin(theta)
for small theta
acceleration = g/l (theta) sin(theta)= theta

Answer 26

u know wave...
omega= 2 pi (nu)

Answer 27

yep

Answer 28

so we say SHM is a periodic motion in which acceleration is directed towards a fixed point which is called equilibrium point.

Answer 29

ok. I know the basics of it

Answer 30

ok..
so acceleration = g/l (theta)

Answer 31

for small theta

Answer 32

yep

Answer 33

now compare this with
a= - w^2 x
where x is the displacement from fixed point

Answer 34

so for small angle x = theta

Answer 35

so acceleration and displacement are in opposite direction

Answer 36

yes seems logical

Answer 37

here theta is x....
for small angle
sin(theta)= theta

Answer 38

so in the case of pendulum
angular displacement and angular acceleration are in opposite direction

Answer 39

write six in the form a power series
sin x= x - x^3/3! + x^5/5!-------

Answer 40

so for small x
sin x =x

Answer 41

neglect higher order terms

Answer 42

aha

Answer 43

so we have
acceleration =- g/l (theta)

Answer 44

yes

Answer 45

so u have w^2= g/l

Answer 46

so T= 2 pi/w

Answer 47

I forgot what w is equal to

Answer 48

w= angular frequency= 2 pi( frequency)

Answer 49

so T = 2pi/2pi?

Answer 50

T= (2 pi)/ (2 pi( frequency))