Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define and calculate Potential Energy (PE) and Kinetic Energy (KE).
• State and apply the principle of conservation of mechanical energy.
• Explain the difference between conservative and non-conservative forces.
• Solve problems using the conservation of mechanical energy.
• Describe energy transformations in simple mechanical systems.

Lesson summary

This week, we delve into the fascinating world of energy, specifically mechanical energy. Understanding energy is crucial because it’s the driving force behind everything that moves or changes around us. In South Africa, this knowledge is especially relevant. From understanding how a solar geyser heats water to the mechanics of a minibus taxi accelerating, energy principles are at play. We'll explore how energy transforms and, most importantly, how it is conserved. This understanding forms the foundation for more advanced topics in physics and engineering.

Lesson notes

2.1 Potential Energy (Gravitational) Potential energy (PE) is the energy an object possesses due to its position or condition. Gravitational potential energy is specifically the energy an object has because of its height above a reference point (usually the ground). The formula for gravitational potential energy is: PE = mgh where: PE is the potential energy (measured in Joules, J) m is the mass of the object (measured in kilograms, kg) g is the acceleration due to gravity (approximately 9.8 m/s² on Earth, but can be rounded to 10 m/s² for easier calculations in some cases) h is the height of the object above the reference point (measured in meters, m)

Example 1: A bag of mielie meal with a mass of 5 kg is placed on a shelf that is 2 meters above the floor. What is the gravitational potential energy of the mielie meal relative to the floor?

Solution: PE = mgh PE = (5 kg)(9.8 m/s²)(2 m) PE = 98 J The mielie meal has 98 Joules of gravitational potential energy. This means it has the potential to do 98 Joules of work if it were to fall to the floor. 2.2 Kinetic Energy Kinetic energy (KE) is the energy an object possesses due to its motion.

The formula for kinetic energy is: KE = ½mv² where: KE 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)

Example 2: A minibus taxi with a mass of 1500 kg is travelling at a speed of 20 m/s (approximately 72 km/h). What is its kinetic energy?

Solution: KE = ½mv² KE = ½(1500 kg)(20 m/s)² KE = ½(1500 kg)(400 m²/s²) KE = 300,000 J KE = 300 kJ The minibus taxi has 300,000 Joules (or 300 kiloJoules) of kinetic energy. This is the energy of motion. 2.3 Mechanical Energy Mechanical energy (ME) is the sum of the potential energy and kinetic energy of an object. ME = PE + KE 2.4 Conservation of Mechanical Energy The principle of conservation of mechanical energy states that in a closed system where only conservative forces are acting, the total mechanical energy remains constant. In other words, energy can be transformed from potential to kinetic, or vice versa, but the total amount of mechanical energy stays the same. A conservative force is one where the work done by the force is independent of the path taken. Gravity is a conservative force. Mathematically, this can be expressed as: ME initial = ME final PE initial + KE initial = PE final + KE final Example 3: A soccer ball with a mass of 0.45 kg is dropped from a height of 5 meters. Assuming air resistance is negligible (meaning we only have gravity, a conservative force), what is the speed of the ball just before it hits the ground?

Solution: Initial state: The ball is at rest at a height of 5 meters. KE initial = 0 J, PE initial = mgh = (0.45 kg)(9.8 m/s²)(5 m) = 22.05 J Final state: The ball is just about to hit the ground, h = 0 m. PE final = 0

J. We need to find KE final = ½mv². Using the conservation of mechanical energy: ME initial = ME final PE initial + KE initial = PE final + KE final 22.05 J + 0 J = 0 J + ½(0.45 kg)v² 22.05 J = 0.225 kg * v² v² = 22.05 J / 0.225 kg v² = 98 m²/s² v = √(98 m²/s²) v ≈ 9.9 m/s The speed of the soccer ball just before it hits the ground is approximately 9.9 m/s. 2.5 Non-Conservative Forces Non-conservative forces are forces where the work done does depend on the path taken. Friction and air resistance are examples of non-conservative forces. When non-conservative forces are present, mechanical energy is not conserved. Some of the mechanical energy is converted into other forms of energy, such as heat (due to friction). In these cases, ME initial ≠ ME final . Instead, some energy is "lost" due to the work done by the non-conservative force. While we won't quantitatively deal with non-conservative forces this week, it's important to understand their effect on energy conservation. Guided Practice (With Solutions)

Question 1: A stone with a mass of 0.2 kg is thrown vertically upwards with an initial velocity of 15 m/s. What is the maximum height the stone will reach, assuming air resistance is negligible?

Solution: Initial State: KE initial = ½mv² = ½(0.2 kg)(15 m/s)² = 22.5

J. PE initial = 0 J (assuming the ground is our reference point).

Final State: At the maximum height, the stone's velocity is 0 m/s. KE final = 0

J. We need to find the height, h, so PE final = mgh.

Using conservation of mechanical energy: ME initial = ME final KE initial + PE initial = KE final + PE final 22.5 J + 0 J = 0 J + (0.2 kg)(9.8 m/s²)h 22.5 J = (1.96 N)h h = 22.5 J / 1.96 N h ≈ 11.48 m Therefore, the maximum height the stone will reach is approximately 11.48 meters.

Question 2: A child slides down a frictionless slide. The slide is 3 meters high. If the child starts from rest at the top of the slide, what will be their speed at the bottom? (Assume g = 9.8 m/s²)

Solution: Initial State: The child is at rest at the top of the slide (h = 3 m). KE initial = 0 J.