A simple harmonic oscillator consists of a block of mass 1.20 kg attached to a spring of spring constant 190 N/m. When t = 1.80 s, the position and velocity of the block are x = 0.124 m and v = 3.190 m/s. (a) What is the amplitude of the oscillations? What were the (b) position and (c) velocity of the block at t = 0 s?
w=sqrtk/m=12.58rad/s
how do I do the rest using a calcultor without a rad function? only asic tring functions?
To solve the problem using a calculator without a radian function, you can convert the angular frequency (w) from radians to degrees and then perform the calculations using trigonometric functions available on your calculator.
Here's how you can calculate the amplitude:
1. Calculate the period (T) of the oscillations using the formula T = 2π/w, where w is the angular frequency. Use the angular frequency you calculated: w = 12.58 rad/s.
T = 2π / 12.58 = 0.499 s
2. The period (T) is related to the frequency (f) by the equation f = 1/T. Calculate the frequency:
f = 1 / 0.499 = 2.006 Hz
3. The amplitude (A) is related to the maximum displacement (x_max) of the oscillations using the formula: x_max = A. To find the maximum displacement, use the position value given at t = 1.80s: x = 0.124 m.
x_max = x = 0.124 m
Therefore, the amplitude of the oscillations is 0.124 m.
Now, let's calculate the position and velocity at t = 0 s:
To find the position at t = 0 s, we need to determine the initial phase (ϕ) of the oscillations. The phase can be found using the equation x = A * cos(wt + ϕ), where x is the position, A is the amplitude, w is the angular frequency, and t is the time.
Since we know the position (x) and amplitude (A), we can rearrange the equation to solve for the phase:
cos(wt + ϕ) = x / A
cos(0 + ϕ) = x / A (substituting t = 0)
cos(ϕ) = x / A
Let's calculate ϕ using a calculator:
ϕ = cos^(-1) (x / A)
ϕ = cos^(-1) (0.124 / 0.124) = cos^(-1)(1) = 0 radians or 0 degrees
Therefore, the position at t = 0 s is x = A * cos(wt + ϕ) = 0.124 * cos(0 + 0) = 0.124 m.
To find the velocity at t = 0 s, we differentiate the position equation with respect to time:
v = -A * w * sin(wt + ϕ)
At t = 0 s, sin(0 + 0) = 0, so the velocity is v = -A * w * sin(0) = 0.
Therefore, the velocity at t = 0 s is v = 0 m/s.