A 51kg object connected to a spring of 31N/m spring constant oscillates on a horizontal, frictionless surface with amplitude of 5cm. Find the speed of the object when when its position is 3cm.
What on earth is Tamucc?
He meant 'Physics' as the subject instead of 'Tamucc', which stands for Texas A&M University - Corpus Christi. He must be taking the summer physics class at TAMUCC.
is the answer found here
To find the speed of the object when its position is 3cm, we need to use the equation for the motion of a mass-spring system. The equation for the displacement of an object in simple harmonic motion is given by:
x = A * cos(ωt)
Where:
x = displacement of the object from its equilibrium position
A = amplitude of the motion
ω = angular frequency of the motion
t = time
From the given information, we know that the amplitude (A) of the motion is 5cm. We also know the position (x) of the object, which is 3cm. Let's proceed step by step to find the angular frequency (ω) and then the velocity (v) of the object.
Step 1: Find the angular frequency (ω):
The angular frequency (ω) can be obtained from the relation:
ω = √(k / m)
Where:
k = spring constant
m = mass
Given that the spring constant (k) is 31 N/m and the mass (m) is 51 kg, we can substitute these values into the equation to find ω.
ω = √(31 N/m / 51 kg)
ω = √(31 / 51) (N/m/kg)
ω ≈ 1.18 rad/s
Step 2: Find the time (t) from the given displacement (x):
To find the time (t) when the object is at the position of 3cm, we can set up the equation as follows:
x = A * cos(ωt)
Substituting the given values:
3cm = 5cm * cos(1.18 rad/s * t)
Now we need to solve this equation for time (t).
cos⁻¹(3/5) = 1.18 rad/s * t
Using a calculator, we find:
t ≈ 0.553 s
Step 3: Find the velocity (v) at the given position (x):
The velocity (v) of the object at any given time can be obtained by taking the derivative of the displacement equation with respect to time (t):
v = -A * ω * sin(ωt)
Substituting the given values:
v = -5cm * 1.18 rad/s * sin(1.18 rad/s * 0.553 s)
Now we can calculate the velocity (v).
v ≈ -3.26 cm/s
The negative sign indicates that the object is moving in the opposite direction to the positive direction of the x-axis.
Therefore, the speed of the object when its position is 3cm is approximately 3.26 cm/s.