Experimentally it is found that a 6 kg weight stretches a certain spring 6 cm. If the weight is
pulled 4 cm below the equilibrium position and released:
a. Set up the differential equation and associated conditions describing the motion.
b. Find the position of the weight as a function of time.
c. Find the amplitude, period, and frequency of motion
d. Determine the position, velocity and acceleration of the weight 0.5s after it has been
released.
To answer these questions, we need to apply Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position.
Let's go step by step to find the answers:
a. To set up the differential equation, we will use Newton's second law of motion, which relates force, mass, and acceleration. The force exerted by the spring is given by Hooke's Law:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
From Newton's second law, we know that force is also equal to mass times acceleration:
F = m * a
Where m is the mass of the object and a is its acceleration.
Equating the two expressions for force, we have:
m * a = -kx
This is our differential equation that describes the motion of the weight.
b. To find the position of the weight as a function of time, we need to solve the differential equation. This can be done by assuming a solution of the form:
x(t) = A * cos(ωt + φ)
Where x(t) is the position of the weight at time t, A is the amplitude, ω is the angular frequency, and φ is the phase constant.
By differentiating x(t) with respect to time, we can find the velocity and acceleration as functions of time.
x'(t) = -A * ω * sin(ωt + φ)
x''(t) = -A * ω^2 * cos(ωt + φ)
c. To find the amplitude, period, and frequency of motion, we can use the given information that the 6 kg weight stretches the spring by 6 cm.
Since the weight is pulled 4 cm below the equilibrium position, the amplitude is 4 cm (or 0.04 m).
The period of motion is the time taken for one complete cycle of motion. It can be calculated using the formula:
T = (2π) / ω
Where T is the period and ω is the angular frequency.
The frequency of motion is the reciprocal of the period, given by:
f = 1 / T
d. To determine the position, velocity, and acceleration of the weight 0.5s after it has been released, we can substitute the values into the expressions we derived for x(t), x'(t), and x''(t) above.