At time t=0 an object of mass 50.34 g on the end of a horizontal spring is moving to the right at 10 cm/s and is displaced 2 cm to the left from the equilibrium position. If the spring constant is 0.234N/m, find an expression for the position of the particle as a function of time.
To find an expression for the position of the particle as a function of time, we need to use the equation of motion for a mass-spring system.
The equation of motion for a mass-spring system is given by:
m * d²x/dt² = -k * x
where:
m = mass of the object (in kg)
x = displacement from the equilibrium position (in meters)
t = time (in seconds)
k = spring constant (in N/m)
In this case, we are given:
m = 50.34 g = 0.05034 kg
k = 0.234 N/m
x = -2 cm = -0.02 m
We also need to find the initial velocity (v₀) of the particle at time t=0. Since the object is moving to the right at 10 cm/s, the initial velocity v₀ is 10 cm/s = 0.1 m/s.
Now, let's solve the equation of motion to find the position of the particle as a function of time:
m * d²x/dt² = -k * x
Substituting the given values:
0.05034 * d²x/dt² = -0.234 * (-0.02)
Simplifying:
d²x/dt² = 0.234 * 0.02 / 0.05034
d²x/dt² = 0.093 / 0.05034
d²x/dt² ≈ 1.846
Now, we have a second-order linear differential equation. To solve it and find x as a function of t, we'll assume a solution of the form:
x(t) = A * cos(ωt) + B * sin(ωt)
where:
A and B are constants to be determined
ω = sqrt(k / m) is the angular frequency
By taking the first and second derivatives of x(t), we can substitute them back into the equation of motion to find the values of A and B:
First derivative:
dx/dt = -A * ω * sin(ωt) + B * ω * cos(ωt)
Second derivative:
d²x/dt² = -A * ω² * cos(ωt) - B * ω² * sin(ωt)
Substituting these derivatives back into the equation of motion:
m * (-A * ω² * cos(ωt) - B * ω² * sin(ωt)) = -k * (A * cos(ωt) + B * sin(ωt))
Simplifying:
-A * ω² * m * cos(ωt) - B * ω² * m * sin(ωt) = -k * A * cos(ωt) - k * B * sin(ωt)
Comparing coefficients of sin and cos separately, we have:
-A * ω² * m = -k * B (1)
-B * ω² * m = -k * A (2)
From equations (1) and (2), we can solve for A and B:
A = - (ω² * m) / k * B (3)
Substituting equation (3) into equation (2):
B * ω² * m = -k * (- (ω² * m) / k * B)
Simplifying:
ω² * m = (ω² * m) / k
1 = 1
From this, we can see that the equation holds true, so there is no restriction on the values of A and B.
Therefore, the expression for the position of the particle as a function of time is:
x(t) = A * cos(ωt) + B * sin(ωt)
where A and B can take any values, and
ω = sqrt(k / m) is the angular frequency.