A 53.6 kg circus performer bobs up and down at the end of a long elastic rope at a rate of once every 2.80 s. The elastic rope obeys Hooke's law. Find how much the rope is extended beyond its unloaded length when the performer hangs at rest.
** I think Im missing a step when i do this problem. Help would be greatly appreciated**
To find how much the rope is extended beyond its unloaded length, we can use Hooke's Law, which states that the force exerted by an elastic material is directly proportional to the displacement of the material from its equilibrium position.
Let's begin by identifying the key information given in the problem:
- Mass of the circus performer = 53.6 kg
- Period of oscillation = 2.80 s
Now, let's break down the problem into steps to find the answer:
Step 1: Find the weight of the circus performer.
The weight can be calculated using the formula:
Weight = Mass × Gravitational acceleration (g)
where the gravitational acceleration is approximately 9.8 m/s^2.
Weight = 53.6 kg × 9.8 m/s^2
Weight = 525.28 N
Step 2: Find the force acting on the rope.
The force acting on the rope is equal to the weight of the performer. This is because when the performer is hanging at rest, the force of gravity is balanced by the elastic force in the rope.
Force = 525.28 N
Step 3: Find the spring constant.
Hooke's Law states that the force exerted by an elastic material is directly proportional to the displacement from its equilibrium position. The equation can be written as:
Force = Spring constant × Displacement
In this case, the displacement is the extension of the rope beyond its unloaded length. To find the spring constant, we need to rearrange the equation:
Spring constant = Force / Displacement
Step 4: Find the displacement.
The displacement can be found using the formula for simple harmonic motion:
Displacement = (2π × amplitude) / period
Since there is no mention of the amplitude in the problem, we can assume it to be half of the total displacement. So, the displacement can be calculated as follows:
Displacement = (2π × Amplitude) / Period
Displacement = (2π × Amplitude) / 2πf
Displacement = Amplitude / f
In this case, f is the frequency, which is the reciprocal of the period.
Frequency (f) = 1 / Period
Frequency = 1 / 2.80 s
Now, substitute the values obtained into the equation for the displacement:
Displacement = Amplitude / (1 / 2.80 s)
Displacement = Amplitude × 2.80 s
Step 5: Calculate the spring constant.
Substitute the known values into the equation for the spring constant:
Spring constant = Force / Displacement
Spring constant = 525.28 N / (Amplitude × 2.80 s)
Step 6: Find the extension of the rope.
In Hooke's Law, the displacement is considered positive when the material is stretched. Therefore, the extension of the rope can be found using the equation:
Extension = Force / Spring constant
Extension = 525.28 N / Spring constant
To find the extension, we need to know the value of the spring constant. Unfortunately, the problem does not provide any information about it. This missing information prevents us from calculating the extension accurately.
To solve this problem, we would need more data about the rope or the properties of the material used, such as the spring constant or elastic modulus. Without that information, it is not possible to determine the extension beyond the unloaded length.