A body of mass 5.0 kg is suspended by a spring which stretches 10 cm when the mass is attached. It is then displaced downward an additional 5.0 cm and released. Its position as a function of time is approximately?

The spring constant is

k = 5.0kg*9.8 m/s^2/0.10 m = 490 N/m
The oscillation angular frequency is
w = sqrt (k/M) = 9.90 rad/s

The mass vibrates with an amplitude of 5 cm about the equilibrium position (10 cm stretch). Measured about that position, the deflection (positive downward) is
Y = 5.0 cm * cos (wt)= 5 cos (9.90 t)

Well, this situation is no laughing matter, but I'll do my best to humor you with an answer. When you have a mass-spring system, the motion is approximately simple harmonic motion (SHM). In SHM, the position of the object as a function of time can be described by a sine or cosine function. So, the position of the body as a function of time in this case would be something like:

x(t) = A * cos(ωt + φ)

Where A is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant. In this case, we need to find the values of A, ω, and φ.

First, let's find the values of A and ω. The amplitude A is equal to the maximum displacement, which is the additional 5.0 cm the body was displaced downward, added to the 10 cm the spring initially stretched. A = 10 cm + 5.0 cm = 15.0 cm.

The angular frequency ω can be found using the formula ω = sqrt(k/m), where k is the spring constant and m is the mass. We don't have the spring constant given, so it's impossible to determine ω without it. And without ω, we can't accurately describe the position of the body as a function of time.

So, unfortunately, I can't provide you with a specific answer in this case. However, I hope I was still able to bring a smile to your face with my attempt at humor!

To find the position of the body as a function of time, we need to consider the motion of the mass-spring system.

1. Determine the spring constant, k:
The spring constant can be calculated using Hooke's Law:
F = k * x,
where F is the force exerted by the spring, k is the spring constant, and x is the displacement.
Given that the spring stretches 10 cm when the mass is attached, we can set up the equation as:
F = k * 0.1 m,
where 0.1 m is the displacement in meters.
The force exerted by the spring is equal to the gravitational force, which can be calculated as:
F = m * g,
where m is the mass and g is the acceleration due to gravity (assumed to be 9.8 m/s^2).
Therefore, we have:
m * g = k * 0.1 m,
5.0 kg * 9.8 m/s^2 = k * 0.1 m,
k = (5.0 kg * 9.8 m/s^2) / 0.1 m.

2. Determine the angular frequency, ω:
The angular frequency can be calculated using the equation:
ω = √(k / m),
where k is the spring constant and m is the mass.
Substituting the values, we have:
ω = √((5.0 kg * 9.8 m/s^2) / 0.1 m * 5.0 kg).

3. Determine the period, T:
The period can be calculated using the equation:
T = (2π) / ω,
where ω is the angular frequency.
Substituting the value of ω, we have:
T = (2π) / √((5.0 kg * 9.8 m/s^2) / 0.1 m * 5.0 kg).

4. Determine the position function, x(t):
The position of the mass-spring system as a function of time can be represented as:
x(t) = A * cos(ωt + φ),
where A is the amplitude, ω is the angular frequency, and φ is the phase constant.
Since the mass is initially displaced downward an additional 5.0 cm (0.05 m), the amplitude (A) is equal to 0.1 m + 0.05 m = 0.15 m.
The phase constant (φ) is assumed to be 0 in this case.

Therefore, the position function, x(t), is given by:
x(t) = 0.15 m * cos(√((5.0 kg * 9.8 m/s^2) / 0.1 m * 5.0 kg) * t).

This equation gives the position of the mass as a function of time.

To find the position of the body as a function of time, we need to use the equation that describes the motion of a mass-spring system. The equation is given by:

x(t) = A * cos(ωt + φ) + x₀

where:
- x(t) is the position of the body as a function of time,
- A is the amplitude of the motion,
- ω is the angular frequency,
- t is time,
- φ is the phase constant, and
- x₀ is the equilibrium position of the body.

In this case, since the body is initially displaced downward by an additional 5.0 cm, the equilibrium position (x₀) is 10 cm + 5.0 cm = 15 cm.

To determine the values of A, ω, and φ, we need to know the details of the spring-mass system. In particular, we need to know the spring constant (k) of the spring.

Once we have the spring constant, we can calculate the values of A, ω, and φ using the following formulas:

A = √(2g/k)
ω = √(k/m)
φ = -π/2

where:
- g is the acceleration due to gravity (approximately 9.8 m/s²), and
- m is the mass of the body.

Without knowing the spring constant (k) or any other details of the system, it is not possible to determine the position of the body as a function of time. Therefore, we cannot provide an exact answer to the question.