Could someone show me how to work this? It's for a study guide I'm practicing, and I just can't seem to get the correct answer.

The position of a 0.63 kg object attached to a spring is described by [x = (1.09 m) cos(3.11πt)]

a) Find the object’s position at t = 1.09 s. Answer in meters.

b) Find the object’s acceleration at the same time. Answer in m/s2.

c) What is the frequency of oscillations in Hz?

d) What is the period of oscillations in s?

I've tried using d=x*t for the first part, but I think I must be missing something. This is more advanced stuff than I'm used to so I'm not sure how to go about it

For part a)

Just substitute in t into the equation you have.
x = (1.09)cos(3.11πt)
x is the displacement and this will come out as an answer in meters. 1.09m is your max amplitute (the number in front of the cos). If you get a negative answer that just means the spring is being compressed so you get a negative displacement

for part b)

I think for this because you have x in terms of t, dx/dt = velocity and dv/dt = acceleration. So if you get d2x/dt2 (the second derivitive) of

(1.09)cos(3.11πt)

and then substitute in the value of t you have for part a) you'll get the acceleration

for part c)

For SHM x = (A)cos(2πt/T)
where t = time and T = period

put 2πt/T = 3.11πt and solve for T

for part d) Frequency = one over the period or

f = 1/T

I'm not getting the right answer for part b. Could someone help me understand this part better?

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.