A spring to which is attached a 2.5kg object is stretched 4cm from its equilibrium position and released to perform simple harmonic motion. The velocity of the object is 2m/s as it passes through the equilibrium position. what is the value of spring constant.
F = -k x = m a
let x = A sin (2 pi f t)
then v = A (2pi f) cos (2 pi f t)
a = -A (2 pi f)^2 sin (2 pi f t)
max extension when sin = 1 or (2 pi f t) = pi/2 , pi/2 + pi, etc
then at max extension A = x = 4 cm = 0.04 meter
then v = 4 (2pi f) cos (2 pi f t)
that v is max when the cos = 1 or -1 etc = 2 m/s
2 = 4 (2 pi f)
so
2 pi f = 0.5
so then
x = .04 sin 0.5 t
v = .04*0.5 cos 0.5 t
a = -.04 (0.25)sin 0.5 t
now back to F = m a
kx = -m a
k (.04 sin .05 t) = - 2.5 * -.04(0.25) sin 0.5 t
k = 2.5 * 0.25
To find the value of the spring constant, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
Hooke's Law can be written as: F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant, and
x is the displacement from the equilibrium position.
Given information:
Mass of the object (m) = 2.5 kg
Displacement from equilibrium position (x) = 4 cm = 0.04 m
Velocity at equilibrium position (v) = 2 m/s
We know that velocity is maximum at the equilibrium position in simple harmonic motion. Therefore, when the object passes through the equilibrium position, its velocity is at its maximum. The maximum velocity can be related to the spring constant and mass by the equation:
v = ωA
Where:
ω is the angular frequency of the motion, and
A is the amplitude of the oscillation.
We can rearrange this equation to solve for the angular frequency (ω):
ω = v / A
Amplitude (A) is half of the total displacement, so A = 0.5 * x.
Substituting the given values, we have:
A = 0.5 * 0.04 = 0.02 m
ω = 2 m/s / 0.02 m = 100 rad/s
Now, we can find the spring constant using the formula:
k = m * ω^2
Substituting the values:
k = 2.5 kg * (100 rad/s)^2
k = 2.5 kg * 10000 rad^2/s^2
k = 25000 N/m
Therefore, the value of the spring constant is 25000 N/m.