Understanding Momentum, Energy, and Circular Motion in Physics

1. Mr. Klapholz Shaker Heights High School Mechanics (2) Momentum, Energy, and Circular Motion. In nearly every physics textbook, this would be at least 3 chapters. For us, it’s half of a chapter.

2. Momentum (p) • p = mv • Momentum is a vector. • Units: kg m s-1 • What is the momentum of a 2 kg object moving westward at 3 m/s? • p = mv = (2)(3) = 6 kg m/s, West

3. Impulse and Momentum • Force is the thing that changes momentum. • The greater the duration of the force, the greater the change in momentum: • Change in Momentum = Force × Time • Definition: “Impulse” = Force × Time • So,… Dp = Impulse F×T= Dp

4. Conservation of Momentum • Change in Momentum = Force × Time • What happens if there is no ‘external’ force acting on a system? • Change in Momentum = 0 × Time • Change in Momentum = 0 • That means the momentum does not change. • There are 7 conserved quantities in nature, and momentum is one of them.

5. Conservation of Momentum • Can the momentum of an object change? • Yes, if a force acts on an object, then the momentum will change. • Can the momentum of a system of objects change? • Yes, if a force acts on the system, then the momentum can change. • But, if the total force that the world puts on a system is zero, then the total momentum of the system does not change.

6. Conservation of Momentum • By far the most common types of momentum problems are ‘explosions’ and ‘collisions’. • In ‘explosions’ and ‘collisions’, the total momentum of the objects stays the same.

7. Momentum vs. Energy • In every collision, momentum is conserved (the total momentum before equals the total momentum after). • In most collisions, energy does not appear to be conserved (the energy seems to decrease). • In one type of collision, even the energy is conserved: “elastic” collisions.

8. Work • Work ≈ Force × Distance • Work is a scalar. • The work done by a particular force depends on how much of the force (Fcosq) is in the direction of the motion. • Work done by a force = {component of force in the direction of motion} × Distance • W = F × s × cosq • Units: Newtons × meters = Joules

9. Example • Calculate the work done by a 100 N force if the object moves 10 m. • A) the angle between the force and the motion is 0˚. • B) the angle between the force and the motion is 45˚. • C) the angle between the force and the motion is 90˚.

10. Solution • Work = Fscosq • Work = (100 N)(10 m)cos0˚ = (1000 J) 1 = 1000 J • Work = (100 N)(10 m)cos45˚ = (1000J)(.707)=707J • Work = (100 N)(10 m)cos90˚ = (1000 J) 0 = 0 • As the angle increases, the force is less involved in making the object move. • As the angle increases, the work done by that force decreases

11. Kinetic Energy • Any moving object has energy. • EK = (½)Mv2 • Less commonly: EK = p2/(2M) • Motion matters more than mass.

12. Gravitational Potential Energy • Any object with height has energy. • EP = Mgh (g = 9.8 m/s2) • What matters, is changes in height, and changes in Gravitational Potential Energy: • DEP = MgDh

13. Work and Energy • The energy of a system, or of one object, can change. • Work done on a system increases the energy of the system. • Work done by a system decreases the energy of the system. Work = DEnergy

14. Conservation of Energy • By far, the most common type of energy problem is when a system does not have energy added to it (or energy taken away). • For this kind of system, Work = 0. • Since Work = DEnergy, and Work = 0 in this case, … • 0 = DEnergy • So, if no work is done by (or on) a system, the total energy stays the same.

15. Efficiency • In a perfect world, if you you drag a cart up a ramp, you would have done exactly as much work as if you had lifted it directly up to the top. • In the real world, friction makes the work we need to do greater than theoretical value. Efficiency = Ideal value ÷ Actual Value (memorize) • If the theoretical work required was 100 J, and it actually required 200 J, then what is the efficiency? • What is the greatest efficiency possible?

16. Power • The rate at which energy changes from one form to another, is ‘power’. • Power = Energy ÷ Time • Power = Force × velocity • Units: Joules per second = Watts = W

17. Circular Motion Intro (1 of 2) • Without exception, if an object is moving along a circular path, then the object is changing direction. • Since its direction is changing, its velocity is changing. • Since its velocity is changing, the object is accelerating (even if the speed is constant).

18. Circular Motion Intro (2 of 2) • Acceleration requires a force. • That force is always toward the center. • That force can be due to a string, gravity, magnetism, electricity, friction, a wall,… but it always called a “centripetal” force. • At its core it is as simple as making a shopping cart turn left. To make circular motion, you must have a force toward the center.

19. Centripetal Acceleration • The acceleration that an object has, due to its change in direction, is called centripetal acceleration. • Greater speed makes greater Centripetal acceleration. • Greater radius makes less Centripetal acceleration. • ac = v2 / R • ac= 4p2R / T2