A ball of mass oscillates on a spring with spring constant = 200 . The ball's position is described by (0.350 ) 16.0 with measured in seconds.
-What is the amplitude of the ball's motion?
-What is the frequency of the ball's motion?
-What is the value of the mass ?
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Ronald Thomas
Part A -
What is the amplitude of the ball's motion?
0.175
0.350
0.700
7.50
16.0
Part B -
What is the frequency of the ball's motion?
0.35
2.55
5.44
6.28
16.0
Part C -
What is the value of the mass ?
0.450
0.781
1.54
3.76
6.33
Part D -
What is the total mechanical energy of the oscillator?
-What is the ball's maximum speed?
Your statement
<<The ball's position is described by (0.350 ) 16.0 with measured in seconds>>
is incomplete. You need to explain which number represents what, and what are the dimensions that go along with your numbers.
Part A - The amplitude of the ball's motion is 0.350.
Part B - The frequency of the ball's motion is 2.55.
Part C - The value of the mass is 0.781.
Part D - To calculate the total mechanical energy of the oscillator, we can use the formula: E = (1/2)kA^2, where E is the total mechanical energy, k is the spring constant, and A is the amplitude. Plugging in the given values, we get: E = (1/2)(200)(0.350)^2 = 12.25.
To find the ball's maximum speed, we can use the formula: vmax = Aω, where vmax is the maximum speed, A is the amplitude, and ω is the angular frequency. The angular frequency can be calculated using the formula: ω = 2πf, where f is the frequency. Plugging in the given values, we get: ω = 2π(2.55) ≈ 16.03 rad/s. Therefore, vmax = (0.350)(16.03) ≈ 5.61. The ball's maximum speed is approximately 5.61.
To find the answers to these questions, we need to understand the equations and principles behind the motion of a mass-spring system.
A mass-spring system can be described by the equation:
x(t) = A * sin(ωt + φ)
Where:
x(t) is the position of the mass at time t,
A is the amplitude of motion,
ω is the angular frequency,
t is the time, and
φ is the phase constant.
Now, let's find the answers to each question:
Part A - What is the amplitude of the ball's motion?
In the equation above, the amplitude (A) is the maximum displacement from the equilibrium position. From the given information, we can see that the maximum displacement is 0.350 m, so the amplitude of the ball's motion is 0.350 m.
Part B - What is the frequency of the ball's motion?
The frequency (f) is related to the angular frequency (ω) by the equation:
f = ω / (2π)
From the given information, we know that the angular frequency (ω) is 16.0 s^-1. Plugging this value into the equation above, we can calculate the frequency as follows:
f = 16.0 / (2π) ≈ 2.55 Hz
So the frequency of the ball's motion is approximately 2.55 Hz.
Part C - What is the value of the mass?
To find the value of the mass (m), we need to use the equation of motion for a mass-spring system:
m * d^2x / dt^2 + k * x = 0
Given that the spring constant (k) is 200 N/m and the angular frequency (ω) is 16.0 s^-1, we can use the equation:
ω = √(k / m)
From this equation, we can solve for the mass (m) as follows:
m = k / ω^2
= 200 / (16.0^2)
≈ 0.781 kg
So the value of the mass is approximately 0.781 kg.
Part D - What is the total mechanical energy of the oscillator?
The total mechanical energy (E) of the oscillator is the sum of the kinetic energy (KE) and the potential energy (PE).
For a mass-spring system, the kinetic energy is given by the equation:
KE = (1/2) * m * v^2
The potential energy is given by the equation:
PE = (1/2) * k * x^2
Given that the amplitude (A) is 0.350 m, we can find the maximum speed (v) of the ball using the equation:
v = ω * A
Substituting the values into the equations above, we can find the total mechanical energy as follows:
E = KE + PE
= (1/2) * m * v^2 + (1/2) * k * x^2
Unfortunately, without more information about the system's initial conditions or the time, we cannot determine the total mechanical energy of the oscillator or the ball's maximum speed.