A 50-g mass is attached to a spring and undergoes simple harmonic motion. Its maximum acceleration is 15m/s2 and its maximum speed is 3.5m/s .

Determine the angular frequency.
Determine the spring constant.
Determine the amplitude of the motion.

how do I find these things?

50 g = .050 kg = m

sinusoidal motion kinematics:
x = A sin wt
v = Aw cos wt
a = -Aw^2 sin wt = -w^2x

givens:
A w^2 = 15
A w = 3.5
so
3.5 w = 15
w = 15/3.5 radians /second first answer

Newton #2
F = -kx = m a
so
-k x = -m w^2 x
or
w^2 = k/m
w = sqrt (k/m)
15/3.5 = sqrt (k/.050)
solve for k - second answer

A = amplitude = 3.5/w third answer

To find the angular frequency, spring constant, and amplitude of the motion, we can use the formulas related to simple harmonic motion.

1. Angular Frequency:
The angular frequency (ω) can be found using the formula:
ω = √(k/m)
where k is the spring constant and m is the mass.

2. Spring Constant:
The spring constant (k) can be found using the formula:
k = m × ω²
where m is the mass and ω is the angular frequency.

3. Amplitude:
The amplitude (A) of the motion can be found using the formula:
A = vmax / ω
where vmax is the maximum velocity and ω is the angular frequency.

Now, let's calculate these values one by one:

Given:
Maximum acceleration (amax) = 15 m/s²
Maximum speed (vmax) = 3.5 m/s
Mass (m) = 50 g = 0.05 kg

1. Angular Frequency:
Using the formula ω = √(k/m), we need to find ω. Since the maximum acceleration is given, we can relate it to the angular frequency using the formula amax = ω² × A.
ω² = amax / A
Substituting amax = 15 m/s² and A = 3.5 m/s, we get:
ω² = 15 m/s² / (3.5 m/s) = 4.2857 rad/s²
Taking the square root of this value, we find:
ω = √4.2857 rad/s² ≈ 2.07 rad/s

2. Spring Constant:
Using the formula k = m × ω², we can substitute the known values:
k = 0.05 kg × (2.07 rad/s)² = 0.216 N/m

3. Amplitude:
Using the formula A = vmax / ω, we can substitute the known values:
A = 3.5 m/s / (2.07 rad/s) ≈ 1.69 m

Therefore, the angular frequency is approximately 2.07 rad/s, the spring constant is 0.216 N/m, and the amplitude of the motion is approximately 1.69 m.

To find the angular frequency, spring constant, and amplitude of the motion, you can use the following equations:

1. Angular frequency (ω): ω = 2πf, where f is the frequency of the motion.
2. Frequency (f): f = 1/T, where T is the period of the motion.
3. Period (T): T = 2π/ω.
4. Spring constant (k): k = mω^2, where m is the mass attached to the spring.
5. Amplitude (A): A = vmax/ω, where vmax is the maximum speed of the motion.

Let's calculate each of these values step by step:

1. Angular frequency (ω):
We need the frequency of the motion first, which we can find from the maximum speed of the mass. The maximum speed occurs when the displacement is at its maximum value (A). From the equation vmax = ωA, we can rearrange it to get ω = vmax / A.

ω = 3.5 m/s / A

2. Frequency (f):
Now that we have the angular frequency, we can find the frequency by using the formula f = ω / 2π.

f = (3.5 m/s / A) / (2π)

3. Period (T):
Using the frequency, we can find the period of the motion by using the formula T = 1 / f.

T = 1 / (3.5 m/s / A) / (2π)

4. Spring constant (k):
To find the spring constant, we need the mass (m) attached to the spring. The formula for the spring constant is k = mω^2.

k = (50 g) * (3.5 m/s / A)^2

5. Amplitude (A):
To find the amplitude, we need the maximum speed (vmax) and the angular frequency (ω). The equation for the amplitude is A = vmax / ω.

A = 3.5 m/s / (3.5 m/s / A)

Simplify the equation A = 3.5 m/s / (3.5 m/s / A) to get A = A. This tells us that the amplitude can be any value since it cancels out.

So, to summarize, to find the angular frequency (ω), divide the maximum speed (3.5 m/s) by the amplitude (A). To find the frequency (f), divide the angular frequency by 2π. To find the period (T), divide 1 by the frequency. To find the spring constant (k), multiply the mass (50 g) by the square of the angular frequency. And finally, the amplitude (A) can be any value.