# Simple harmonic motion spring question

1.
A spring with spring constant 14.5 n/m hangs from the ceiling.
A 430g ball is attached to the spring and allowed to come to rest.
It is then pulled down 7.20cm and released.

What is the time constant if the ball's amplitude has decreased to 3.70 after 52.0 oscillations?

## The Attempt at a Solution

w = squareroot (14.5 n/m / 0.43kg)

vmax = 2pi * 3.7 / T
vmax = w*a
w*a = 2pi*3.7/ T
solve for T

i know it's damping

not sure what it means by the time constant, nor what formula to use to solve for it

help would be appreciated, and an explanation of how you got to the awnser

tiny-tim
Homework Helper
… not sure what it means by the time constant, nor what formula to use to solve for it

Hi vorcil!

An exponential process A(t) = A0e-λt has decay constant λ and time constant (or mean lifetime) 1/λ

Hi vorcil!

An exponential process A(t) = A0e-λt has decay constant λ and time constant (or mean lifetime) 1/λ

so 3.7 cm = 7.2cm * e^ (lavender t)

how do i get lavender??????
and what am i suppose to be solving for, is it t?

or because lavender = 1/t do i go e^ (lavender * 1/lavender)
and solve for lavender, then solve for t??????????

tiny-tim
Homework Helper
how do i get lavender??????

erm … it's spelt (and pronounced) lambda (originally, I think, from the Hebrew word for "swan")!

(and why didn't you just copy-and-paste my λ? )

You get λ by using A(t) = A0e-λt, and plugging in the data given to you, and solving.

erm … it's spelt (and pronounced) lambda (originally, I think, from the Hebrew word for "swan")!

(and why didn't you just copy-and-paste my λ? )

You get λ by using A(t) = A0e-λt, and plugging in the data given to you, and solving.

what is the t in e-λt, i would be able to solve for lavender if i had that little t,

and then once i got lavender i'd just use 1/lavender to solve for the time constant am i correct?

my first physics teacher was asian so i always thought he was pronouncing it lavender lol, lambda XD lamb duh

Last edited:
what is the t in e-λt, i would be able to solve for lavender if i had that little t,

and then once i got lavender i'd just use 1/lavender to solve for the time constant am i correct?

my first physics teacher was asian so i always thought he was pronouncing it lavender lol, lambda XD lamb duh

w= squareroot spring constant / mass

w = 2pi / t
2pi / w = t

3.2 = 7.2 * e^(-x*1.08)
got lavender being 0.61
1/0.61 = 1.63, awnser was wrong

I tried using the energy approach but got an unsolvable awnser

here:

tried using the energy way, i need to find out how much time it takes for the energy in the system to damp by 37%
that's what Tau is, t

E$$_{}o$$ * e ^ -t / $$\tau$$ = E(t) = 1/2 k xmax^2 (3.2cm)

omfg i did it, I SOLVED IT BOOYAH, my method

the intital amplitude at t = 0 is A = 7.2 cm
the time taken to reach 1/2 A is aquired by the following
w = squareroot k/mass

T = 2pi / w
2pi/5.806 = 1.08 seconds
and that is for each oscillation

now to reach 52 oscillations it takes 52 * 1.08 = 56.27 seconds
and that is the point which it reaches 3.7 cm

now solving for HALF the amplitude, 7.2cm / 2 = 3.6cm NOT 3.7 cm
so i have to find out how many oscillations it will have to do for 3.6 cms,
so if 7.2 - 3.7 = 3.5

3.5 /52 = 0.06730769231 is the distance reduced by each oscillation
7.2 - 3.6 = 3.6
3.6 / 06730769231 = 53.48 = is the number of oscillations needed to reach 3.6 cm

now i multiply that by the time it takes for each oscillation (from above)
53.48 * 1.08 = 57.88 seconds is the time needed to dampen to half the amplitude

from the formulas
1/2 A = A * e^(-t/2 tau)

the A's cancel out and tau is the time constant

1/2 = e^(-t / 2 tau )

take the logs
ln(1/2) = ln e ^ (-t / 2tau)

ln(1/2) = -ln2
-ln2 = t / 2tau
-ln2 * 2tau = t
t/2ln2 = tau

57.88 / 2ln2 = tau
41.75!! AND IT WAS CORRECT
except off my a couple sigfigs so i must've rounded somewhere badlly
right awnser was 42.3 but it still gave me the correct marking

tiny-tim
Homework Helper
Hi vorcil!

(just got up :zzz: …)

Well done for perservering, and even more so for getting it right!

Here's some extra symbols for you to copy-and-paste …

(and also try using the X2 tag just above the Reply box )