Question

You pick up a six-pack of soft drinks from the floor, but it slips from your hand and smashes onto your foot. Comment on the work and energy involved in this sequence. What forms of energy are involved at what stages of the process?

Short answer

Work converts potential energy to kinetic energy, then to thermal, sound, and mechanical energy upon impact.

Step-by-step solution

Initial State - Lifting the Six-Pack

Initially, you exert an upward force to lift the six-pack from the floor to a certain height. As you apply force and move the six-pack upward, you do work on it. The work done (W) is calculated using the formula: \[ W = F imes d imes ext{cos}( heta) \] Where F is the force applied, d is the displacement in the direction of force, and \( \theta \) is the angle between the force and displacement (which is zero in this case, as both are upwards). At this stage, the gravitational potential energy (GPE) of the six-pack increases as it gains height. The GPE is given by: \[ ext{GPE} = m imes g imes h \] where m is the mass, g is the acceleration due to gravity, and h is the height.

Transition - Slipping

When the six-pack slips from your hand, the potential energy begins converting into kinetic energy because the six-pack starts to fall under the influence of gravity. At the moment it begins to slip, the gravitational potential energy is at its maximum and kinetic energy is zero. As it falls, potential energy decreases while kinetic energy increases.

Impact - Collision with the Foot

When the six-pack collides with your foot, its kinetic energy is at its maximum, and the potential energy is at its minimum (the potential energy is zero at the ground level). The kinetic energy is converted into other forms of energy: 1. Thermal energy due to deformation of the cans and impact. 2. Sound energy when the impact generates noise. 3. Mechanical energy causing potential damage to the foot or the cans.

Key concepts

Gravitational Potential Energy

When you lift an object from the ground, like the six-pack in this scenario, you are giving it gravitational potential energy (GPE). This energy is due to the object's position relative to the Earth. The formula to calculate GPE is \( \text{GPE} = m \times g \times h \), where:

• \(m\) is the mass of the object in kilograms.
• \(g\) is the acceleration due to gravity, approximately \(9.8\, \text{m/s}^2\) on Earth.
• \(h\) is the height in meters above the ground.

As you lift the six-pack, its height from the ground increases and so does its GPE. This energy remains stored until the six-pack starts moving downwards, just like a book held at a height or a rock positioned at the top of a hill. Understanding GPE helps us see how energy conservation works and how potential shifts into other energy forms.

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. When the six-pack slips from your hand, it gains speed and thus kinetic energy as it starts to fall. The formula to calculate kinetic energy is \( KE = \frac{1}{2} m v^2 \), where:

• \(m\) is the mass of the object.
• \(v\) is the velocity of the object.

In the beginning, when you hold the six-pack and it's not moving, its kinetic energy is zero. But as soon as it starts to fall, gravity pulls it down, accelerating the six-pack and increasing its velocity. This acceleration changes the potential energy it had while being lifted into kinetic energy. Understanding kinetic energy is crucial as it explains why objects gain energy while moving and what happens to this energy during collisions.

Energy Transformation

Energy transformation refers to the process of changing one form of energy into another. In the case of the six-pack falling, several transformations take place:

• Initially, when you lift the six-pack, you transform chemical energy (from your muscles) into gravitational potential energy.
• As it begins to fall, gravitational potential energy transforms into kinetic energy.
• Upon impact with your foot, kinetic energy transforms into other forms, such as thermal energy, sound energy, and mechanical energy.

These transformations illustrate the law of conservation of energy, which states that energy cannot be created or destroyed, only transformed. This principle is foundational in physics as it allows us to predict the outcome of different energy interactions in everyday life.