An oscillator with a mass of 281 g has a speed of 119.1 cm/s when its displacement is 2.80 cm and 34.2 cm/s when its displacement is 7.80 cm. What is the oscillator's maximum speed (in m/s)?
Conservation of energy tells you that:
(1/2)k(2.80)^2 + (1/2)m(119.1)^2 =
(1/2)k(7.80)^2 + (1/2)m(34.2)^2
m (the mass) is known; solve for the spring constant, k.
Then compute the maximum energy, which is the sum of the terms on either side.
That will equal (1/2)*M*Vmax^2
Finally, solve for Vmax
To find the oscillator's maximum speed, we need to understand the relationship between speed and displacement in simple harmonic motion.
In simple harmonic motion, the relationship between speed (v) and displacement (x) is given by the equation:
v = ω√(A^2 - x^2)
where ω is the angular frequency and A is the amplitude.
In this case, we are given the speed (v) and displacement (x) at two different points in the oscillation. Using this information, we can set up a system of equations and solve for the amplitude (A).
Let's denote the first set of values as (x1, v1) and the second set of values as (x2, v2).
Given:
x1 = 2.80 cm = 0.0280 m
v1 = 119.1 cm/s = 1.191 m/s
x2 = 7.80 cm = 0.0780 m
v2 = 34.2 cm/s = 0.342 m/s
Substituting the values into the equation, we get two equations:
v1 = ω√(A^2 - x1^2)
v2 = ω√(A^2 - x2^2)
Squaring both sides of the equations to eliminate the square root, we have:
v1^2 = ω^2(A^2 - x1^2)
v2^2 = ω^2(A^2 - x2^2)
Now, divide the two equations to eliminate ω^2:
v1^2 / v2^2 = (A^2 - x1^2) / (A^2 - x2^2)
Substituting the given values:
(1.191 m/s)^2 / (0.342 m/s)^2 = (A^2 - (0.0280 m)^2) / (A^2 - (0.0780 m)^2)
Solving this equation will give us the amplitude (A) of the oscillator.
After finding the value of A, we can calculate the maximum speed (v_max) using the equation:
v_max = ωA
where ω is the angular frequency and we can find it using the equation:
ω = 2πf
where f is the frequency. However, we are not given the frequency in this problem to directly solve for ω. Therefore, we can use the equation:
ω = 2π / T
where T is the period of the oscillation. We can find the period (T) using the equation:
T = 1 / f
Now, we have all the necessary information to solve for the maximum speed (v_max).