Today: Work, Kinetic Energy, Work-Energy Theorem

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Motivation: Why do we learn about work and energy???

Energy : Central concept in science and engineering … and in our daily lives…

Energy transforms from one type to other, but conserved.

In Physics 102, we learn about “Mechanical Energy”.

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1 Kinetic energy Kinetic energy (K.E.): An energy associated with motion

Kinetic energy of an object of mass m and speed v

1 v KE..== K mv2 2 m Æ Energy is a scalar quantity (No direction!)

SI Unit for energy: J (Joule) mm2 Jkg=⋅ =⋅⋅=⋅ kgmNm ss22

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1 K..EK== mv2 2

The heavier and the faster, the kinetic energy gets greater.

(KE of a fast baseball) vs (KE of a slow baseball) ?

(KE of a car at 60 mi/h) vs (KE of a trailer at 60 mi/h) ?

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2 iClicker

When the velocity of the object doubles, its kinetic energy ______.

(a) doubles (b) quadruples (c) becomes half (d) unchanged

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Example A baseball outfielder throws a baseball of mass 0.15 kg at a speed of 40 m/s and initial angle of 30 degree. What is the kinetic energy of the baseball at the highest point of the trajectory? Ignore air friction.

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3 What makes kinetic energy change?

(1 D motion)

FxΔ Fnet = ma net

defined as “Net Work”, W net , in 1D Change in kinetic energy is equal to the “net work”! Work-Energy Theorem K −=Δ=KKW f inet7

Definition of work by a single force, F, in 1D:

WFxF = Δ m

(Work) = (Force) x (Displacement) F

Work done by a force F on an object, the displacement of which is Δ x .

If F and Δ x are in the same direction, work done by F is positive.

If F and Δ x are in the opposite direction, work done by F is negative.

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4 iClicker Quiz 1

Work done by a gravity on a falling elephant WFx=Δ is ______.

a) Positive b) Negative c) zero If force and displacement point the same direction, the work is ______.

Example: In this case, suppose m=1000 kg and distance = 10 m, calculate work done by gravity.

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iClicker Quiz2: Work done by a gravity on a ball that is going up

Gravity WFx= Δ

Displacement Is work positive, negative, zero? a) Positive b) Negative c) zero Gravity

If force and displacement point the opposite directions, the work is ______. Example: In this case, suppose m=10 kg and distance = 10 m, calculate work done by gravity. 10

5 Net Work when multiple forces are exerted on a single object

Wnet = Fnet Δx

= (F1 + F2 + F3 +...)Δx

= F1Δx + F2Δx + F3Δx +...

= W1 +W2 +W3 +...

Work-Energy Theorem

K f −=Δ=KKWinet

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Work-Energy Theorem

As shown for 1D motion,

K f −=Δ=KKWinet

Æ Work-Energy Theorem

Positive net work: KE increases (iClicker Quiz 1)

Negative net work: KE decreases (iClicker Quiz 2)

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6 iClicker Quiz 1. As a sled is pulled by dogs across a flat, snow-covered field at a constant velocity, work done by the air resistance and friction is ______,

2. …and the work done by dogs is ______, 3. … and net work done on the sled is ______.

a) Positive b) Zero c) Negative

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Example

A 5-kg object is moving at 7 m/s. A 2-N force is applied in the opposite direction of motion and then removed after the object has traveled an additional 20 m.

A) What is the initial kinetic energy? B) What is the work done by 2N force? C) What is the final kinetic energy?

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7 Example A horizontal force of 200 N is applied to a 55-kg cart across a 10-m rough level surface. If the cart accelerates at 2.0 m/s2, then what is the work done by the force of friction as it acts to retard the motion of the cart?

What is the work done by the 200 N force? What is the net force? Suppose the initial kinetic energy is 300 J. What is the final kinetic energy?

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iClicker

A horizontal force of +100 N is applied in the x direction to a 20-kg cart initially moving at -10 m/s in the x-direction. The cart finally reaches a velocity of zero, reverses direction, and reaches +10 m/s finally.

What is the work done by 100 N force between initial and the time when the velocity reaches zero?

What is the work done by the 100 N force between initial and final? a) -1000 J b) +1000 J c) +2000 J d) -2000 J e) zero

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8 For 2D motion…

Kinetic energy

11222 KE..== mv m() vx + vy 22

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Work in 2D

What if force and displacement are perpendicular?

Example: Uniform circular motion

G G F Fnet net

G Fnet

No change in “magnitude” of velocity Æ No kinetic energy change Æ No work, sorry ! (Velocity does change, because the “direction” changes.)

9 Displacement

positive work

Force

Displacement

zero work

Force

Displacement negative work

Force

Work vs. Disp.-Force angle Cos!

Force and Displacement with an angle

10 Work done by a constant force in 2D K G Define G G W = F d cos θ F , d ≡ F ⋅ d

Scalar (dot) product magnitude Force

θ Displacement

positive work Note: 090≤<θ θ = 90 zero work 90<≤θ 180 negative work 21

Math Review: Scalar (dot) product using components K G G G WFd=≡⋅cosθ Fd, Fd

=+Fdxx Fd yy G G F = (,FF ) x y ddd= (,x y ) G G G G G G G G G G G A⋅B=B⋅A (A + B)⋅C = A⋅C + B ⋅C

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11 iClicker Quiz G ForceFN=−(3 , 2 N ) is acting on an object, G which makes a displacement dm= (2 , 4m)

Find the work done by this force.

A) (6JJ , - 8 )

22 22 B) 32+× 24 + C) 6 + 12 -4 -8 = 6 J D) -2 J E) Not enough information. I need to know theta.

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