An object with mass 3.7 is attached to a spring with spring stiffness constant 270 and is executing simple harmonic motion. When the object is 2.5×10−2 from its equilibrium position, it is moving with a speed of 0.45 . what is the amplitude of the motion?
ω=sqrt(k/m)
v= ω•sqrt(A²-x²)
To find the amplitude of the motion, we need to use the equations that describe simple harmonic motion (SHM).
The equation that relates the displacement (x) of an object in SHM to its amplitude (A) and its position at any given time is:
x = A * cos(ωt)
where ω is the angular frequency given by:
ω = √(k/m)
k is the spring constant (also known as spring stiffness constant) and m is the mass of the object.
Given:
Mass (m) = 3.7 kg
Spring constant (k) = 270 N/m
Displacement (x) = 2.5 × 10^(-2) m
Velocity (v) = 0.45 m/s
First, let's find the angular frequency (ω) using the formula ω = √(k/m):
ω = √(270 / 3.7) ≈ 5.867 rad/s
Now, we will use the given displacement (x = 2.5 × 10^(-2) m) to find the amplitude (A) using the following equation:
x = A * cos(ωt)
When the object is at its maximum displacement (either positive or negative), the cosine function will have a value of 1. Therefore, we can write:
A = x / cos(ωt)
Now, we need to find the value of the cosine function. To do that, we will use the given velocity (v = 0.45 m/s). At the maximum displacement, the velocity is zero since the object momentarily stops before changing direction.
v = -A * ω * sin(ωt)
Since v = 0 at the maximum displacement, we have:
0 = -A * ω * sin(ωt)
sin(ωt) = 0, which means ωt = 0 or ωt = π (radians)
Since the cosine function is the reciprocal of the sine function (cos(θ) = 1/sin(θ)), we can substitute ωt = 0 or ωt = π into cos(ωt) to find the value of the cosine function:
cos(0) = 1 or cos(π) = -1
Since the sign of the cosine function doesn't affect the amplitude, we can choose either 1 or -1.
Let's substitute the values into A = x / cos(ωt):
A = (2.5 × 10^(-2) m) / 1
A ≈ 2.5 × 10^(-2) m
Therefore, the amplitude of the motion is approximately 2.5 × 10^(-2) meters.