Question

Physics Question: Mechanics
Circular Motion
A stone is rotated steadily in a horizontal circle with a period T by a string of length l. if the tension in the string is constant and l increases by 1% find the percentage change in T.

Answer 1

The answer should 0.5%

Answer 2

in circular motion, F = mv^2/r

Answer 3

woops , T = 2pi*r/v

Answer 4

here the centripetal force is provided by the tension in the string, so
Tension = m*v^2/r

Answer 5

ok do i have to use ratios at any time?

Answer 6

you dont have to, if you can find another way

Answer 7

can you show me how to solve it then please thanks

Answer 8

since Force is constant we have

Answer 9

m*v^2/ r = m *x^2 / (1.01*r) , because the radius increased by 1% , solve for x, the new speed

Answer 10

ok

Answer 11

did you get x = sqrt(1.01)*v ?

Answer 12

yep

Answer 13

ok great, now plug this into the T (period) equation

Answer 14

T = 2pi * r / v

Answer 15

plug this 'new' speed in for v

Answer 16

yes

Answer 17

ok

Answer 18

Let's call the new period T '. The old period, with the old radius and old speed was T = 2pi * r / v

The new speed is sqrt(1.01)*v
The new radius is 1.01*r
Plugging in we have

T ' = 2*pi * (1.01*r) / ( sqrt(1.01)* v )
= (2pi * r /v ) * 1.01 / sqrt( 1.01)
= T * 1.01 / sqrt(1.01)
= T * 1.004987
= T ( 1 + .005) approximately
= T( 1 + .5% )

Answer 19

@Michele_Laino can you double check this is correct

Answer 20

i was copying your earlier posts and some seem to have been deleted

Answer 21

the crucial point to notice here is that if F = mv^2 / r , and F is constant (since F is tension and it is given that tension is constant), then if you increase radius then you must increase velocity in order for the fraction to stay the same.

Think about a fraction, if i want to increase the bottom, i have to increase the top as well if i want keep the fraction the same number overall

Answer 22

or you could increase mass, but in the problem mass is fixed , so that forces velocity to change

Answer 23

yes thats clear

Answer 24

ok

Answer 25

so what we had initially was the equation:

Tension = mv^2 / r = m v ' ^2 / r ' , since tension stayed constant.

we were given that r ' = 1.01 * r
and we solved for v ' in terms of v

Answer 26

and you see how we got the period T equation ( T does not stand for tension but for time of period)

Answer 27

we know that average speed = distance / time ,

In a circle, since we have constant speed we can choose distance to be a one revolution , and T is the time it takes to make one revolution (also known as period).
v = circumference / T
v = 2pi*r / T

solve for T
T = 2pi * r / v

Answer 28

I hope that helped :)

Answer 29

it did clearest explanation i have had so far thanks a lot

Answer 30

also you should check to see if the answer makes intuitive sense. The answer we got is that the period increased by 0.5% when we increased radius by 1% but kept tension fixed. Does this make sense?
Yes because there will be a larger circumference to cover so it makes sense that period (time to complete 1 revolution) increases.