Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define and calculate Gravitational Potential Energy (GPE) and Kinetic Energy (KE).
• State and apply the principle of conservation of mechanical energy.
• Explain the effect of non-conservative forces like friction.
• Differentiate between conservative and non-conservative forces.

Lesson summary

This week, we delve into the fascinating world of energy, specifically focusing on mechanical energy and the principle of its conservation. Understanding energy is crucial for understanding how the world around us works. From the electricity powering our homes to the cars we drive, energy is the driving force. In South Africa, with our need for sustainable energy solutions and our involvement in various industrial and engineering projects, grasping these concepts is not just academic but vital for future problem-solving and innovation. The efficient use of resources and development of renewable energy sources depend on this foundation.

Lesson notes

2.1 Energy: The Ability to Do Work Energy is defined as the ability to do work. The SI unit for energy is the Joule (J). We will focus on two types of mechanical energy: Kinetic Energy (KE): The energy an object possesses due to its motion. It depends on the object's mass (m) and velocity (v).

The formula for kinetic energy is: ``` KE = 1/2 m v^2 ``` Where: KE is kinetic energy (measured in Joules, J) m is mass (measured in kilograms, kg) v is velocity (measured in meters per second, m/s)

Example: A taxi with a mass of 1500 kg is travelling at 20 m/s on the N1 highway. What is its kinetic energy? ``` KE = 1/2 m v^2 KE = 1/2 1500 kg (20 m/s)^2 KE = 1/2 1500 kg 400 m^2/s^2 KE = 300,000 J ``` Gravitational Potential Energy (GPE): The energy an object possesses due to its position relative to a reference point (usually the ground). It depends on the object's mass (m), the acceleration due to gravity (g), and the object's height (h) above the reference point. On Earth, g is approximately 9.8 m/s². The formula for gravitational potential energy is: ``` GPE = m g h ``` Where: GPE is gravitational potential energy (measured in Joules, J) m is mass (measured in kilograms, kg) g is the acceleration due to gravity (approximately 9.8 m/s²) h is height (measured in meters, m)

Example: A bag of mealie meal with a mass of 5 kg is placed on a shelf 1.5 meters above the ground. What is its gravitational potential energy relative to the ground? ``` GPE = m g h GPE = 5 kg 9.8 m/s² 1.5 m GPE = 73.5 J ``` 2.2 Conservation of Mechanical Energy In a closed system where only conservative forces (like gravity) are acting, the total mechanical energy remains constant. Mechanical energy is the sum of kinetic energy and gravitational potential energy: ``` Mechanical Energy (ME) = KE + GPE ``` The principle of conservation of mechanical energy states: ``` ME_initial = ME_final KE_initial + GPE_initial = KE_final + GPE_final ``` This means that energy can be transformed from one form to another (e.g., from potential to kinetic) but the total amount of mechanical energy stays the same.

Important Assumptions: This law holds true only when: Friction is negligible (e.g., a block sliding on ice). Air resistance is negligible. No external forces (other than gravity) are doing work on the system.

Example: A ball with a mass of 0.2 kg is dropped from a height of 10 meters. Assuming no air resistance, what is its velocity just before it hits the ground?

Initial State: KE_initial = 0 J (since the ball is initially at rest) GPE_initial = m g h = 0.2 kg 9.8 m/s² * 10 m = 19.6 J Final State (just before hitting the ground): GPE_final = 0 J (since the height is 0) KE_final = 1/2 m v^2 = 1/2 0.2 kg * v^2 Applying Conservation of Mechanical Energy: ``` KE_initial + GPE_initial = KE_final + GPE_final 0 J + 19.6 J = 1/2 0.2 kg v^2 + 0 J 19.6 J = 0.1 kg * v^2 v^2 = 19.6 J / 0.1 kg v^2 = 196 m^2/s^2 v = √(196 m^2/s^2) v = 14 m/s ``` Therefore, the velocity of the ball just before it hits the ground is 14 m/s. 2.3 Non-Conservative Forces and Work Done Conservative Forces: Forces for which the work done is independent of the path taken. Gravity is a conservative force. If you lift an object straight up or move it along a winding path to the same height, the change in GPE is the same.

Non-Conservative Forces: Forces for which the work done does depend on the path taken. Friction and air resistance are non-conservative forces. If you slide a box across a rough floor, the longer the distance you slide it, the more work friction does, and the more energy is lost as heat. When non-conservative forces are present, the total mechanical energy is not conserved. The work done by these forces results in a change in mechanical energy: ``` W_non-conservative = ΔME = ME_final - ME_initial W_non-conservative = (KE_final + GPE_final) - (KE_initial + GPE_initial) ```

Example: A box with a mass of 2 kg slides down a ramp that is 3 meters long and 1 meter high. The coefficient of kinetic friction between the box and the ramp is 0.

2. What is the velocity of the box at the bottom of the ramp?

Initial State: KE_initial = 0 J GPE_initial = m g h = 2 kg 9.8 m/s² * 1 m = 19.6 J Final State: GPE_final = 0 J KE_final = 1/2 m v^2 = 1/2 2 kg * v^2 = v^2 Work Done by Friction: First we need to calculate the Normal force: F_N = mgcos(θ). We need to figure out what theta is. sin(θ) = 1m/3m. θ = arcsin(1/3) = 19.47 degrees. F_N = 2kg9.8m/s^2 * cos(19.47) = 18.42 N Frictional force F_f = μF_N = 0.2 18.42 N = 3.684 N W_friction = F_f d cos(180) = 3.684 N 3 m * (-1) = -11.05 J (negative because friction opposes motion).

Applying the Work-Energy Theorem: ``` W_non-conservative = ΔME = ME_final - ME_initial -11.05 J = (v^2 + 0 J) - (0 J + 19.6 J) -11.05 J = v^2 - 19.6 J v^2 = 19.6 J - 11.05 J v^2 = 8.55 m^2/s^2 v = √8.55 m^2/s^2 v = 2.92 m/s ``` Therefore, the velocity of the box at the bottom of the ramp is 2.92 m/s.