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Work & Energy WORK Force x Distance WORK Force x Distance Energy

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Presentation on theme: "Work & Energy WORK Force x Distance WORK Force x Distance Energy"β€” Presentation transcript:

1 Work & Energy WORK Force x Distance WORK Force x Distance Energy
Kinetic + (Potential)

2 Work and Energy Intro example Chapter 6 Roadmap Method Differences
Crate example 2 important points about work

3 Chapter 6 review Start with equation 2-11c 𝑣 2 = 𝑣 π‘œ 2 +2π‘Žπ‘₯
𝑣 2 = 𝑣 π‘œ 2 +2π‘Žπ‘₯ v2 and vo2 are scalars e.g π‘š 𝑠 2 = βˆ’2 π‘š 𝑠 2 So 2ax must be scalar If a and x in same direction, larger v2 If a and x in opposite direction, smaller v2 If a and x perpendicular, same v2

4 Scalar product of 2 vectors
Could define scalar product of a and x π‘Žβˆ™π‘₯= π‘Ž π‘₯ π‘π‘œπ‘ πœƒ Result If a and x in same direction, 2ax positive, larger v2 If a and x in opposite direction, 2ax negative, smaller v2 If a and x perpendicular, 2ax zero, same v2

5 Work and energy Equation 2-11c Multiply by Β½ m 𝑣 2 = 𝑣 π‘œ 2 +2π‘Žπ‘₯
𝑣 2 = 𝑣 π‘œ 2 +2π‘Žπ‘₯ Multiply by Β½ m 1 2 π‘š 𝑣 2 = π‘šπ‘£ π‘œ 2 +π‘šπ‘Žπ‘₯ 1 2 π‘š 𝑣 2 = π‘šπ‘£ π‘œ 2 +𝐹π‘₯π‘π‘œπ‘ πœƒ New kinetic energy Old kinetic energy Work

6 π‘Šπ‘œπ‘Ÿπ‘˜ = 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 πΈπ‘›π‘’π‘Ÿπ‘”π‘¦ π‘“π‘–π‘›π‘Žπ‘™ βˆ’ 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 πΈπ‘›π‘’π‘Ÿπ‘”π‘¦ π‘–π‘›π‘–π‘‘π‘–π‘Žπ‘™
Work and Energy 𝐹 βˆ†π‘₯ cos πœƒ = 1 2 π‘šπ‘£ 2 βˆ’ π‘š 𝑣 π‘œ 2 π‘Šπ‘œπ‘Ÿπ‘˜ = 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 πΈπ‘›π‘’π‘Ÿπ‘”π‘¦ π‘“π‘–π‘›π‘Žπ‘™ βˆ’ 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 πΈπ‘›π‘’π‘Ÿπ‘”π‘¦ π‘–π‘›π‘–π‘‘π‘–π‘Žπ‘™ Work equals change in Kinetic Energy All scalars, use only magnitudes! Units N-m, or kg m2/s2 Joules (J)

7 Generalizing equation 2-11c
Modified 3rd Equation 2π‘Ž βˆ†π‘₯ cos πœƒ = 𝑣 2 βˆ’ 𝑣 π‘œ 2 Consider several cases 𝐼𝑓 𝒂 𝑖𝑛𝑙𝑖𝑛𝑒 π‘€π‘–π‘‘β„Ž βˆ†π’™, 𝑙𝑒𝑓𝑑 𝑠𝑖𝑑𝑒 π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’, 𝑣 π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘  𝐼𝑓 𝒂 π‘Žπ‘™π‘šπ‘œπ‘ π‘‘ 𝑖𝑛𝑙𝑖𝑛𝑒 π‘€π‘–π‘‘β„Ž βˆ†π’™, 𝑙𝑒𝑓𝑑 𝑠𝑖𝑑𝑒 π‘ π‘œπ‘šπ‘’π‘€β„Žπ‘Žπ‘‘ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’, 𝑣 π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘  π‘Ž 𝑙𝑖𝑑𝑑𝑙𝑒 𝐼𝑓 𝒂 π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ π‘‘π‘œ βˆ†π’™, 𝑙𝑒𝑓𝑑 𝑠𝑖𝑑𝑒 π‘§π‘’π‘Ÿπ‘œ, 𝑣 π‘Ÿπ‘’π‘šπ‘Žπ‘–π‘›π‘  π‘ π‘Žπ‘šπ‘’ 𝐼𝑓 𝒂 π‘Žπ‘™π‘šπ‘œπ‘ π‘‘ π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ βˆ†π’™, 𝑙𝑒𝑓𝑑 𝑠𝑖𝑑𝑒 π‘ π‘œπ‘šπ‘’π‘€β„Žπ‘Žπ‘‘ π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’, 𝑣 π‘‘π‘’π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘  π‘Ž 𝑙𝑖𝑑𝑑𝑙𝑒 𝐼𝑓 𝒂 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ βˆ†π’™, 𝑙𝑒𝑓𝑑 𝑠𝑖𝑑𝑒 π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’, 𝑣 π‘‘π‘’π‘π‘Ÿπ‘’π‘Žπ‘ π‘’π‘  Product of a and Ξ”x, and how they’re working together, either increases/decreases/keeps-constant v2 Note v2 is scalar, no direction!

8 Conclusions Product of force, distance, and how they’re working together increases or decreases the magnitude of v. How force and distance work together is very important. If f and d inline, magnitude of v increases. If f and d partially inline, magnitude of v increases a little. If f and d perpendicular, magnitude of v remains constant. If f and d partially opposed, magnitude of v decreases a little. If f and d opposed, magnitude of v decreases. If f but no d v remains constant.

9 Method Differences Chapter 3 Chapter 4 Chapter 5 Chapter 6
Position, velocity, acceleration vectors. X and y components. Chapter 4 Force and acceleration vectors. Ξ£F = ma is vector equation. Solve F=ma in x and y directions. Chapter 5 Solve F=ma in radial and other directions. Chapter 6 Work and energy scalars. Forget direction, throw everything in β€œbig mixing pot”.

10 Future Roadmap Combination of Force, Distance, and how they’re working together creates scalar WORK. WORK either increases or decrease scalar KINETIC ENERGY – involves velocity magnitude. Some types of WORK are always difference of 2 endpoints, and can be treated as difference in scalar POTENTIAL ENERGY. LOSS OF PE often equals GAIN OF KE (or vice-versa). Thus POTENTIAL + KINETIC (scalar) is CONSERVED Great shortcut – Solve complicated paths looking only at endpoints!

11 Work Definition F . x . cos(ΞΈ) +1 when together Definition
Cos(ΞΈ) extracts F and x working together +1 when together -1 when opposed -1 to +1 when in between 0 when perpendicular Work is a scalar quantity F x

12 Work done by Crate Example 6.1 Method 1 Method 2
50 kg crate, pulled 40 m FP = 100 N, Ffric = 50 N Method 1 Solve for net force 100 N cos(37) – 50 N = 30 N Multiply by 40 m = 1200 J Method 2 Find individual works Wmg = 0, WFn = 0, WFP = 3200, WFfric = -2000 0J + 0 J J – 2000 J = 1200 J Work of sum = sum of works

13 Problem 8 Man lowering piano
Forces Fg = 3234 N Ffric = ΞΌ mg cosΞΈ = 1142 N FP = mg sinΞΈ - ΞΌ mg cosΞΈ = 376 N Works Wfr = 1142 N x 3.6 m (-1) = J WP = 376 N x 3.6 m (-1) = J Wg = 3234 N x (3.6 sin28) = J Wnormal = 0 (perpendicular) Total work is 0 Work of gravity was Fg times height Had it accelerated work would not be 0

14 Problem 8 – work done by gravity (1)
Force component in direction of displacement Using angle between force and displacement - Ο΄ π‘Šπ‘œπ‘Ÿπ‘˜=mgcos⁑(πœƒ)βˆ™π‘‘ displacement mg Ο΄ mg cosΟ΄

15 Problem 8 – work done by gravity (2)
Force component in direction of displacement Using complimentary angle Ξ¦ π‘Šπ‘œπ‘Ÿπ‘˜=mg𝑠𝑖𝑛⁑(πœ™)βˆ™π‘‘ displacement mg Ξ¦ mg sinΞ¦

16 Problem 8 – work done by gravity (3)
Displacement component in direction of force Another way of looking at sin(Ξ¦) π‘Šπ‘œπ‘Ÿπ‘˜=mgβˆ™π‘‘π‘ π‘–π‘›β‘(πœ™) Same thing! So the work done by gravity is just mgh displacement mg Ξ¦ d sinΞ¦

17 Two important things Total Work is Each Individual Work
The work of the sum of all forces Ξ£Fi x distance or The sum of the individual works of all forces. Ξ£(Fi x distancei) Each Individual Work Force component in direction of displacement. Displacement component in direction of force.

18 Work equals change in Kinetic Energy
Work and Energy πΉβˆ™π‘₯ cos πœƒ = 1 2 π‘š 𝑣 𝑓 2 βˆ’ 1 2 π‘š 𝑣 𝑖 2 π‘Šπ‘œπ‘Ÿπ‘˜=βˆ†πΎπΈ Work equals change in Kinetic Energy

19 Work equals change in energy
Work and Energy Fx cosΟ΄ = Β½ mv2 - Β½ mvo2 Work = Ξ”Energy Work equals change in energy

20 Examples of Work and Energy
Example 6.7 – Falling rock Use 2nd law Use work Car going down ramp Example Roller coaster Couldn’t do easily by 2nd law! Vertical circle example (use work) Note how you β€œmix up” dimensions!

21 More Examples of Work and Energy
Example 6.6 – Work to increase car speed Example 6.5 – Work to stop car Problem Air resistance on baseball

22 October Potter County hiking / camping
Outta here October Potter County hiking / camping


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