A 0.125 kg mass is attached to a spring and undergoes simple harmonic motion with a pe- riod of 0.41 s. The total energy of the system is 2.5 J.
Find the amplitude of the motion. Answer in units of m.
Why was the math book sad? Because it had too many problems!
To find the amplitude of the motion, we can use the formula for the total energy of a system in simple harmonic motion, which is given by the equation:
E = (1/2)kA²
where E is the total energy, k is the spring constant, and A is the amplitude of the motion.
We're given that the total energy (E) is 2.5 J. We can substitute this into the equation:
2.5 = (1/2)kA²
We also know that the period (T) is 0.41 s. The formula for the period in simple harmonic motion is defined as:
T = 2π√(m/k)
where m is the mass and k is the spring constant. We can rearrange this equation to solve for k:
k = (4π²m)/(T²)
Substituting the given values:
k = (4π² * 0.125)/(0.41)²
Simplifying:
k = 24.6937
Now we can substitute the value of the spring constant (k) into the equation for energy:
2.5 = (1/2)(24.6937)(A²)
Simplifying:
A² = (2.5 * 2)/(24.6937)
A² = 0.2021
Finally, taking the square root of both sides:
A = √(0.2021)
A ≈ 0.4499 m
So, the amplitude of the motion is approximately 0.4499 meters.
To find the amplitude of the motion, we can use the formula for the total energy of a simple harmonic oscillator:
Total energy (E) = (1/2) * k * A^2
Where:
E is the total energy of the system (given as 2.5 J)
k is the spring constant
A is the amplitude of the motion (what we're trying to find)
Rearranging the formula, we can solve for A:
A = sqrt( (2 * E) / (k) )
Since the mass and period are given, we can find the spring constant using the formula:
k = (2 * pi * m) / T^2
Where:
m is the mass of the object (given as 0.125 kg)
T is the period of the oscillation (given as 0.41 s)
Plugging in the values, we get:
k = (2 * pi * 0.125) / (0.41^2)
Simplifying this, we find:
k ≈ 7.620 N/m
Now, we can substitute the values of E and k into the formula for amplitude:
A = sqrt( (2 * 2.5) / (7.620) )
Calculating this, we find:
A ≈ 0.648 m
Therefore, the amplitude of the motion is approximately 0.648 meters.
To find the amplitude of the motion, we can use the formula for the energy of a mass-spring system:
E = (1/2)kA^2.
Where:
E is the total energy of the system,
k is the spring constant, and
A is the amplitude of the motion.
Given:
E = 2.5 J, and
The period T = 0.41 s.
First, we need to find the angular frequency (ω) of the motion.
The formula for the angular frequency (ω) in terms of the period (T) is given by:
ω = (2π)/T.
Substituting the given period T = 0.41 s into the formula, we have:
ω = (2π)/(0.41) ≈ 15.31 rad/s.
Next, we can find the spring constant (k) using the formula:
k = mω^2.
Where:
m is the mass attached to the spring.
Given:
m = 0.125 kg, and
ω = 15.31 rad/s.
Substituting these values into the formula, we have:
k = 0.125 × (15.31)^2.
Now, we can solve for the spring constant (k):
k ≈ 29.541 kg/s^2.
Finally, we can use the formula for the energy of the system to find the amplitude (A):
A = √(2E/k).
Substituting the values of E and k we found earlier, we have:
A = √(2 × 2.5 / 29.541).
Simplifying this expression, we find:
A ≈ √(0.135) ≈ 0.367 m.
Therefore, the amplitude of the motion is approximately 0.367 meters.