If a pendulum has period T and you triple its length, what is its new period in

Question

If a pendulum has period T and you triple its length, what is its new period in terms of T?

If a pendulum has a length L and you want to quadruple its frequency, what should be its length in terms of L?

Suppose a pendulum has a length L and period T on earth. If you take it to a planet where the acceleration of freely falling objects is five times what it is on earth, what should you do to the length to keep the period the same as on earth?

If you do not change the pendulum's length in part c, what is the period on that planet in terms of T?

If a pendulum has a period T and you quadruple the mass of its bob, what happens to the period (in terms of T)?

Explanation / Answer

time period, T=2pi*sqrt(l/g)

a)

if l'=3*l

new time period,

T'=2pi*sqrt(l'/g)

T'=2pi*sqrt(3*l/g)

T'=sqrt(3)*(2pi*sqrt(l/g))

T'=sqrt(3)*T

T'=1.73*T

b)

if frquency, f'=4*f

then,

f'=1/T'

f'=(1/2pi)*sqrt(g/l')

4*f=(1/2pi)*sqrt(g/l')

4*(1/2pi)*sqrt(g/l)=(1/2pi)*sqrt(g/l')

4*sqrt(g/l)=sqrt(g/l')

16/l=1/l'

===> l'=l/16

new length, l'=l/16

c)

if, g'=5*g

T=2pi*sqrt(l/g)

===>

T'=T

2pi*sqrt(l'/g')=2pi*sqrt(l/g)

sqrt(l'/5*g)=sqrt(l/g)

===>

l'/5=l

l'=5*l --->

new length, l'=5*l

d)

T'=2pi*sqrt(l/g')

T'=2pi*sqrt(l/5*g)

T'=sqrt(1/5)*2pi*sqrt(l/g)

T'=0.45*T

new time period, T'=0.45T

e)

if mass of the changes,

time period is same (T)

time period does not depend on the mass of the bob

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