I'm haivng troulbe proving the deriviation of this formula on my own...
could you please help me derive the proof of the formula or maybe show me one thansk...
I need to prove this formula
period of a mass spring system in simple harmonic motion
T = 2 pi square root of (m/k)
period = 2 pi X square root of (mass divided by spring constant)
thanks
F = m a
F = -k x for spring of constant k in N/m
so
m a = - k x
now assume motion of form x = A sin (2 pi f t)
then
dx/dt = velocity = A (2 pi f) cos 2 pi f t)
then
d velocity/dt = a = -A (2 pi f)^2 sin (2 pi f t)
or in other words
a = - x (2 pi f)^2
so
- k x = m [ - x (2 pi f)^2 ]
or
(2 pi f)^2 = k/m
so
2 pi f = sqrt (k/m)
but T = 1/f
so
T = 2 pi sqrt (m/k)
Sure, I'd be happy to help you derive the formula for the period of a mass-spring system in simple harmonic motion.
To start, let's define the variables involved:
- T: period of the motion (time it takes to complete one full oscillation)
- m: mass of the object attached to the spring
- k: spring constant (also known as the force constant or stiffness of the spring)
The motion of a mass-spring system can be described by the differential equation:
m(d^2x/dt^2) + kx = 0
Where x is the displacement of the mass from its equilibrium position.
To solve this differential equation, we assume a solution of the form x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.
Taking the second derivative of x(t) with respect to time gives us:
d^2x/dt^2 = -Aω^2 cos(ωt + φ)
Substituting this into the differential equation, we get:
-mAω^2 cos(ωt + φ) + kA cos(ωt + φ) = 0
Dividing through by A gives:
-mω^2 + k = 0
Rearranging this equation, we get:
ω^2 = k/m
Taking the square root of both sides, we have:
ω = √(k/m)
The angular frequency is related to the period T by the formula ω = 2π/T.
Substituting this into the equation, we get:
2π/T = √(k/m)
Squaring both sides and rearranging, we arrive at the formula for the period:
T = 2π√(m/k)
This formula represents the period of a mass-spring system in simple harmonic motion.
So, to summarize the derivation:
1. Start with the differential equation for a mass-spring system.
2. Assume a specific form for the displacement as a function of time.
3. Substitute the assumed solution into the differential equation.
4. Solve for the angular frequency ω.
5. Relate ω to the period T.
6. Simplify and rearrange to obtain the final formula for the period T.
I hope this helps!