Lesson Notes By Weeks and Term v5 - Grade 10
Mechanics: energy and conservation of mechanical energy – Week 7 focus
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Mechanics: energy and conservation of mechanical energy – Week 7 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Physical Sciences
Class: Grade 10
Term: 3rd Term
Week: 7
Theme: General lesson support
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This week, we delve into the fascinating world of energy, specifically mechanical energy, and the principle of its conservation. Understanding energy is crucial for explaining how things move and interact in our universe. Energy isn't just an abstract concept; it's fundamental to understanding how a car engine works, how electricity is generated at Koeberg Nuclear Power Station, how we can design more efficient buildings to conserve energy, and even how a soccer ball curves in flight.
2.1 Kinetic Energy (Ek): Kinetic energy is the energy an object possesses due to its motion. The faster the object moves, and the more massive it is, the more kinetic energy it has.
The formula for kinetic energy is: Ek = ½ mv² where: Ek is the kinetic energy, measured in Joules (J) m is the mass of the object, measured in kilograms (kg) v is the velocity of the object, measured in meters per second (m/s)
Why this formula makes sense: A heavier object (larger m) requires more energy to reach the same speed, and increasing the speed (v) has a squared effect, meaning doubling the speed quadruples the kinetic energy. 2.2 Gravitational Potential Energy (Ep): Gravitational potential energy is the energy an object possesses due to its position above the Earth's surface (or any other reference point). The higher the object is, and the more massive it is, the more gravitational potential energy it has. The formula for gravitational potential energy is: Ep = mgh where: Ep is the gravitational potential energy, measured in Joules (J) m is the mass of the object, measured in kilograms (kg) g is the acceleration due to gravity, which is approximately 9.8 m/s² on Earth. h is the height of the object above a reference point (often the ground), measured in meters (m).
Why this formula makes sense: A heavier object (larger m) requires more energy to lift to a certain height, and the higher you lift it (larger h), the more energy you need to use, which is then stored as potential energy. 2.3 Mechanical Energy (Em): Mechanical energy is the total energy of a system due to its motion and position. It is the sum of the kinetic energy and the potential energy. Em = Ek + Ep 2.4 The Principle of Conservation of Mechanical Energy: This principle states that in a closed system where only conservative forces are acting (like gravity), the total mechanical energy remains constant. In simpler terms, energy can be transformed from one form (kinetic or potential) to another, but the total amount of energy remains the same.
Mathematically: Em (initial) = Em (final) Ek (initial) + Ep (initial) = Ek (final) + Ep (final)
Important Considerations: A closed system means no energy enters or leaves the system. Conservative forces are forces like gravity where the work done by the force is independent of the path taken. Friction is a non-conservative force because the work done by friction does depend on the path. If non-conservative forces (like friction or air resistance) are present, some mechanical energy is converted into other forms of energy (like heat or sound), and the total mechanical energy is not conserved. We will qualitatively address the effect of non-conservative forces. 2.5 Worked
Examples: Example 1: A soccer ball with a mass of 0.45 kg is kicked upwards with an initial velocity of 15 m/s.
Calculate: (a) The kinetic energy of the ball at the moment it is kicked. (b) The gravitational potential energy of the ball at its highest point. (c) The maximum height the ball reaches (assuming air resistance is negligible).
Solution: (a) Ek = ½ mv² = ½ 0.45 kg (15 m/s)² = 50.625 J (b) At the highest point, the ball momentarily stops moving, so its kinetic energy is zero. Since mechanical energy is conserved (we're neglecting air resistance), all of the initial kinetic energy is converted to gravitational potential energy.
Therefore, Ep (highest point) = 50.625 J. (c) Ep = mgh => h = Ep / (mg) = 50.625 J / (0.45 kg * 9.8 m/s²) = 11.48 meters.
Example 2: A 2 kg textbook falls from a shelf 1.8 m above the floor. Calculate the velocity of the textbook just before it hits the floor (assuming air resistance is negligible).
Solution: Initially, the textbook has only gravitational potential energy: Ep = mgh = 2 kg 9.8 m/s² 1.8 m = 35.28 J. Its kinetic energy is zero. Just before impact, the textbook has negligible gravitational potential energy (height is essentially zero). All the initial potential energy has been converted to kinetic energy.
Therefore, Ek = 35.28 J. Ek = ½ mv² => v² = (2 Ek) / m = (2 35.28 J) / 2 kg = 35.28 (m/s)² v = √(35.28 (m/s)²) = 5.94 m/s.
Example 3: A roller coaster car with a mass of 500 kg starts from rest at the top of a 30 m high hill. Assuming no friction, what is the speed of the car at the bottom of the hill?
Solution: Initial state (top of the hill): Ek = 0 J (starts from rest), Ep = mgh = 500 kg 9.8 m/s² 30 m = 147000 J Final state (bottom of the hill): Ep = 0 J (height is zero), Ek = ½ mv² Using conservation of mechanical energy: Ek(initial) + Ep(initial) = Ek(final) + Ep(final) 0 J + 147000 J = ½ 500 kg v² + 0 J 147000 J = 250 kg * v² v² = 147000 J / 250 kg = 588 (m/s)² v = √(588 (m/s)²) = 24.25 m/s Example 4: A pendulum with a bob of mass 0.5 kg is released from a height of 0.2 m above its lowest point. Calculate the speed of the bob as it passes through the lowest point of its swing.