A woman bungee-jumper of mass 60 kg is attached to an elastic rope of natural length 15 m . The

A woman bungee-jumper of mass 60 kg is attached to an...

A woman bungee-jumper of mass $60 \mathrm{kg}$ is attached to an elastic rope of natural length $15 \mathrm{m} .$ The rope behaves like a spring of spring constant $k=220 \mathrm{Nm}^{-1}$. The other end of the spring is attached to a high bridge. The woman jumps from the bridge. (a) Determine how far below the bridge she falls, before she instantaneously comes to rest. (b) Calculate her acceleration at the position you found in (a). (c) Explain why she will perform $\mathrm{SH} \mathrm{M}$, and find the period of oscillations. (d) The woman will eventually come to rest at a specific distance below the bridge. Calculate this distance. (e) Explain whether her oscillations are underdamped, critically damped or overdamped. (f) The mechanical energy of the woman after she comes to rest is less than the woman's total mechanical energy just before she jumped. Explain what happened to the 'lost' mechanical energy.

A woman bungee-jumper of mass $60 \mathrm{kg}$ is attached to an elastic rope of natural length
$15 \mathrm{m} .$ The rope behaves like a spring of spring constant $k=220 \mathrm{Nm}^{-1}$. The other end of the spring is attached to a high bridge. The woman jumps from the bridge.
(a) Determine how far below the bridge she falls, before she instantaneously comes to rest.
(b) Calculate her acceleration at the position you found in (a).
(c) Explain why she will perform $\mathrm{SH} \mathrm{M}$, and find the period of oscillations.
(d) The woman will eventually come to rest at a specific distance below the bridge. Calculate this distance.
(e) Explain whether her oscillations are underdamped, critically damped or overdamped.
(f) The mechanical energy of the woman after she comes to rest is less than the woman's total mechanical energy just before she jumped. Explain what happened to the 'lost' mechanical energy.

Key Concepts

Simple Harmonic Motion (SHM)

Simple harmonic motion describes a type of oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium. Such systems, including ideal spring-mass arrangements, display a sinusoidal oscillation characterized by a specific period and frequency determined by the system’s physical parameters such as mass and spring constant.

Damping and Energy Dissipation

Damping refers to the process by which energy is gradually lost from an oscillatory system due to non-conservative forces like friction or air resistance. This energy dissipation results in a decrease in the amplitude of the oscillations over time. In analyzing physical systems, it is important to distinguish between underdamped, critically damped, and overdamped responses to predict how quickly and in what manner the system will settle.

Equilibrium Position

The equilibrium position in a spring-mass system is the point where the net force acting on the mass is zero. This happens when the restoring force of the spring exactly balances the gravitational force. Understanding this concept is crucial for determining the static position where the system would eventually settle if no energy were lost.

Elastic Potential Energy (Hooke's Law)

Elastic potential energy is the stored energy in materials that can be deformed under force, such as springs or elastic cords. According to Hooke's law, the force exerted by a spring is proportional to its displacement from its natural length. This energy is quantified as one half the spring constant multiplied by the square of the displacement, and it plays a key role in analyzing oscillatory motion in spring systems.

Gravitational Potential Energy

Gravitational potential energy is the energy associated with an object due to its position in a gravitational field. It is generally expressed as the product of the object's mass, the gravitational acceleration, and the height above a reference point. This energy becomes particularly important in analyzing systems where objects fall or are elevated.

Energy Conservation

This concept refers to the principle that in a closed system, the total energy remains constant though it may change forms. In mechanics, it is used to analyze how gravitational potential energy converts into other forms such as kinetic or elastic potential energy, especially in problems where non-conservative forces (like air resistance) are negligible or considered separately.

Transcript

00:01 A bungee jumper of mass 60 kilograms is attached to an elastic rope of natural lengths 15 meters.

00:08 The rope behaves like a spring of spring constant 220 newtons per meter.

00:16 The other end of the spring is attached to a high bridge.

00:20 The woman jumps from the bridge, so part a determine how far below the bridge she falls before she instantaneously comes to rest.

00:29 So part a.

00:31 So she's going to fall a distance h before she comes to rest.

00:38 So let the zero of the potential energy be that distance h from the bridge.

00:47 So the gravitational potential energy is mgh.

00:55 So all the potential energy when she comes to rest will be converted into potential energy in the elastic rope.

01:03 And since the rope will act like a spring, the total elastic energy is equal to one -half kx squared.

01:15 So x is equal to h minus 15 meters, and it's going to be equal.

01:24 Remember, to the potential energy.

01:29 So knowing this, we can solve for h.

01:34 We know k, we know x, we know m, we know g.

01:40 So solving for h, it will be a quadratic, and you'll get 27 meters or 8 .3 meters.

01:50 The 8 .3 meters isn't going to work because it's smaller than the length of the rope.

01:57 So the height is 27 meters.

02:02 Part b, it determined the acceleration.

02:05 At the position you found in part a.

02:09 So for part b, we have the net force is kx minus mg is equal to ma.

02:21 So we know x is h minus 15...

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