An 85kg person bounces on a bathroom scale having a force constant of 1.50*10^6 N/m. The amplitude of the bounce is 0.200 cm. What is the maximum energy stored in the spring?
I use the formula mg = kx, solved for x, and got 5.55333 * 10^-4.
Then I added the amplitude, .002 m, to get 2.5553*10^-3, which was wrong. I also tried the answer without adding the amplitude, and that was wrong.
Where could I be going wrong? There was some discussion on the forum of my online class saying that the question might have some issues...where is the error here?
Thank you.
ω = sqrt(k/m) = sqrt(1.5e6/85) = 133rad/s
x(t) = 0.002sinωt = (1/500)*sinωt
v(t) = (ω/500)*cosωt
vmax = (ω/500) = 133/500=0.266 m/s
max energy= 1/2 m vmax^2
you know m, and vmax.
check my math.
To find the maximum energy stored in the spring, you need to calculate the maximum displacement of the spring from its equilibrium position. The formula you mentioned, mg = kx, represents the equilibrium position of the spring, where mg is the weight of the person (mass multiplied by the acceleration due to gravity) and k is the force constant of the spring.
However, to find the maximum displacement, you need to consider the amplitude given in the question. The amplitude represents the maximum displacement from the equilibrium position. In this case, the amplitude is given as 0.200 cm, which is equal to 0.002 m.
To find the maximum displacement, you can use the formula for the displacement of a simple harmonic motion given by x = A * sin(ωt), where A is the amplitude and ω (omega) is the angular frequency. Note that the angular frequency is related to the force constant and mass through the equation ω = sqrt(k/m).
First, calculate the angular frequency ω:
ω = sqrt(k/m) = sqrt((1.50*10^6 N/m) / (85 kg))
Plugging in the values, the angular frequency ω is approximately 108.685 radians per second.
Next, find the maximum displacement x:
x = A * sin(ωt) = 0.002 m * sin(108.685 * t)
To find the maximum displacement, you need to find the maximum value of the sine function, which is 1. Therefore,
x = 0.002 m * 1 = 0.002 m
This means the maximum displacement of the spring is indeed 0.002 m (or 2.00 cm), as given in the question.
Finally, to find the maximum energy stored in the spring, you can use the formula for the potential energy of a spring:
E = (1/2) * k * x^2
Plugging in the values, you get:
E = (1/2) * (1.50*10^6 N/m) * (0.002 m)^2
Calculating this expression, you should find the maximum energy stored in the spring to be approximately 600 joules.
If there is still confusion or if the answer does not match the one provided, there may be an issue with the problem itself or the answer options. In such cases, it is best to consult with your instructor or refer to the discussion on the forum you mentioned.