To solve this problem, we'll use the equation for the position of an object attached to a spring:
x = A * cos(ωt).
In this equation, x represents the position of the object, A is the amplitude of the oscillation, ω is the angular frequency, and t is the time.
a) To find the object's position at t = 1.09 s, we substitute the given values into the equation:
x = (1.09 m) * cos(3.11π * 1.09).
To calculate this, you'll need to use a scientific calculator or an online calculator that can handle trigonometric functions. First, multiply 3.11 by π (pi), then multiply the result by 1.09, and finally take the cosine of the result. The final answer will give you the object's position at t = 1.09 s in meters.
b) To find the object's acceleration at the same time, we can differentiate the position equation to get the equation for acceleration:
a = -A * ω^2 * sin(ωt).
In this equation, a represents the acceleration of the object.
To calculate the acceleration, substitute the given values into the equation:
a = -(1.09 m) * (3.11π)^2 * sin(3.11π * 1.09).
Again, you'll need a scientific calculator or an online calculator capable of handling trigonometric functions. First, square 3.11π, then multiply the result by 1.09, and finally take the sine of the result. The final answer will give you the object's acceleration at t = 1.09 s in m/s^2.
c) The frequency of oscillations (f) is related to the angular frequency (ω) by the equation:
f = ω / (2π).
To find the frequency, divide the given angular frequency (3.11π rad/s) by 2π. The final answer will give you the frequency of the oscillations in Hz (Hertz).
d) The period of oscillations (T) is the reciprocal of the frequency (T = 1 / f). To find the period of oscillations, take the reciprocal of the frequency calculated in the previous step. The final answer will give you the period of oscillations in seconds (s).
Remember to be careful with units and use a calculator with trigonometric functions for the mathematical calculations.