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7.4 Work Done by a Varying Force

Work Done by a Varying Force

- Assume that during a very small displacement, Dx,

F is constant - For that displacement, W F Dx
- For all of the intervals,

Work Done by a Varying Force, cont

- Sum approaches a definite value
- Therefore
- (7.7)
- The work done is equal to the area under the

curve!!

Example 7.7 Total Work Done from a Graph (Example

7.4 Text Book)

- The net work done by this force is the area under

the curve - W Area under the Curve
- W AR AT
- W (B)(h) (B)(h)/2 (4m)(5N) (2m)(5N)/2
- W 20J 5J 25 J

Work Done By Multiple Forces

- If more than one force acts on a system and the

system can be modeled as a particle, the total

work done ON the system is the work done by the

net force - (7.8)

Work Done by Multiple Forces, cont.

- If the system cannot be modeled as a particle,

then the total work is equal to the algebraic sum

of the work done by the individual forces

Hookes Law

- The force exerted BY the spring is
- Fs kx (7.9)
- x is the position of the block with respect to

the equilibrium position (x 0) - k is called the spring constant or force constant

and measures the stiffness of the spring (Units

N/m) - This is called Hookes Law

Hookes Law, cont.

- When x is positive (spring is stretched), Fs is

negative - When x is 0 (at the equilibrium position), Fs is

0 - When x is negative (spring is compressed), Fs is

positive

Hookes Law, final

- The force exerted by the spring (Fs ) is always

directed opposite to the displacement from

equilibrium - Fs is called the restoring force
- If the block is released it will oscillate back

and forth between x and x

Work Done by a Spring

- Identify the block as the system
- The work as the block moves from
- xi xmax to xf 0

(7.10)

Work Done by a Spring, cont

- The work as the block moves from
- xi 0 to xf xmax

(7.10) (a)

Work Done by a Spring, final

- Therefore
- Net Work done by the spring force as the block

moves from xmax to xmax is ZERO!!!! - For any arbitrary displacement xi to xf

(7.11)

Spring with an Applied Force

- Suppose an external agent, Fapp, stretches the

spring - The applied force is equal and opposite to the

spring force - Fapp Fs (kx) ? Fapp kx

Spring with an Applied Force, final

- Work done by Fapp
- when xi 0 to xf xmax is
- WFapp ½kx2max
- For any arbitrary displacement xi to xf

(7.12)

Active Figure 7.10

7.5 Kinetic Energy And the Work-Kinetic Energy

Theorem

- Kinetic Energy is the energy of a particle due to

its motion - K ½ mv2 (7.15)
- K is the kinetic energy
- m is the mass of the particle
- v is the speed of the particle
- Units of K Joules (J)
- 1 J Nm (kgm/s2)m kgm2/s2 kg(m/s)2
- A change in kinetic energy is one possible result

of doing work to transfer energy into a system

Kinetic Energy, cont

- Calculating the work
- Knowing that
- F ma mdv/dt m(dv/dt)(dx/dx) ?
- Fdx m(dv/dx)(dx/dt)dx mvdv
- (7.14)

Work-Kinetic Energy Theorem

- The Work-Kinetic Energy Principle states
- SW Kf Ki DK (7.16)
- In the case in which work is done on a system and

the only change in the system is in its speed,

the work done by the net force equals the change

in kinetic energy of the system. - We can also define the kinetic energy
- K ½ mv2 (7.15)

Work-Kinetic Energy Theorem, cont

- Summary Net work done by a constant force in

accelerating an object of mass m from v1 to v2

is - Wnet ½mv22 ½mv12 ? DK
- Net work on an object Change in Kinetic

Energy - Its been shown for a one-dimension constant

force. However, this is valid in general!!!

Work-Kinetic Energy REMARKS!!

- Wnet work done by the net (total) force.
- Wnet is a scalar.
- Wnet can be positive or negative since ?K can be

both or - K ? ½mv2 is always positive. Mass and v2 are

both positive. (Question 10 Homework) - Units are Joules for both work kinetic energy.
- The work-kinetic theorem relates work to a

change in speed of an object, not to a change in

its velocity.

Example 7.8 Question 14

- (a). Ki ? ½m v2 0
- K depends on v2 0 m gt 0
- If v ?2v
- Kf ½m (2v)2 4(½mv2 ) 4Ki
- Then Doubling the speed makes an objects

kinetic energy four times larger - (b). If SW 0 ? v must be the same at the final

point as it was at the initial point

Example 7.9 Work-Kinetic Energy Theorem (Example

7.7 Text Book)

- m 6.0kg first at rest is pulled to the right

with a force F 12N (frictionless). - Find v after m moves 3.0m
- Solution
- The normal and gravitational forces do no work

since they are perpendicular to the direction of

the displacement - W F Dx (12)(3)J 36J
- W DK ½ mvf2 0 ?
- 36J ½(6.0kg)vf2 (3kg)vf2
- Vf (36J/3kg)½ 3.5m/s

Example 7.10 Work to Stop a Car

- Wnet Fdcos180 Fd Fd
- Wnet ?K ½mv22 ½mv12 Fd ?
- -Fd 0 ½m v12 ? d ? v12
- If the cars initial speed doubled, the stopping

distance is 4 times greater. Then d 80 m

Example 7.11 Moving Hammer can do Work on Nail

- A moving hammer strikes a nail and comes to rest.

The hammer exerts a force F on the nail, the nail

exerts a force F on the hammer (Newton's 3rd

Law) m - Work done on the nail is positive
- Wn ?Kn Fd ½mnvn2 0 gt 0
- Work done on the hammer is negative
- Wh ?Kh Fd 0 ½mhvh2 lt 0

Example 7.12 Work on a Car to Increase Kinetic

Energy

- Find Wnet to accelerate the 1000 kg car.
- Wnet ?K K2 K1 ½m v22 ½m v12
- Wnet ½(103kg)(30m/s)2 ½(103kg)(20m/s)2 ?
- Wnet 450,000J 200,000J 2.50x105J

Example 7.13 Work and Kinetic Energy on a Baseball

- A 145-g baseball is thrown so that acquires a

speed of 25m/s. ( Remember v1 0) - Find (a). Its K.
- (b). Wnet on the ball by the pitcher.
- (a). K ? ½mv2 ½(0.145kg)(25m/s)2 ?
- K ? 45.0 J
- (b). Wnet ?K K2 K1 45.0J 0J ?
- Wnet 45.0 J

7.6 Non-isolated System

- A nonisolated system is one that interacts with

or is influenced by its environment - An isolated system would not interact with its

environment - The Work-Kinetic Energy Theorem can be applied to

nonisolated systems

Internal Energy

- The energy associated with an objects

temperature is called its internal energy, Eint - In this example, the surface is the system
- The friction does work and increases the internal

energy of the surface

Active Figure 7.16

Potential Energy

- Potential energy is energy related to the

configuration of a system in which the components

of the system interact by forces - Examples include
- elastic potential energy stored in a spring
- gravitational potential energy
- electrical potential energy

Ways to Transfer Energy Into or Out of A System

- Work transfers by applying a force and causing

a displacement of the point of application of the

force - Mechanical Waves allow a disturbance to

propagate through a medium - Heat is driven by a temperature difference

between two regions in space

More Ways to Transfer Energy Into or Out of A

System

- Matter Transfer matter physically crosses the

boundary of the system, carrying energy with it - Electrical Transmission transfer is by electric

current - Electromagnetic Radiation energy is transferred

by electromagnetic waves

Examples of Ways to Transfer Energy

- d) Matter transfer
- e) Electrical Transmission
- f) Electromagnetic radiation

- a) Work
- b) Mechanical Waves
- c) Heat

Conservation of Energy

- Energy is conserved
- This means that energy cannot be created or

destroyed - If the total amount of energy in a system

changes, it can only be due to the fact that

energy has crossed the boundary of the system by

some method of energy transfer - Mathematically SEsystem ST (7.17)
- Esystem is the total energy of the system
- T is the energy transferred across the system

boundary - Established symbols Twork W and Theat Q
- The Work-Kinetic Energy theorem is a special case

of Conservation of Energy

Material for the Final

- Examples to Read!!!
- Example 7.7 (page 195)
- Example 7.9 (page 201)
- Example 7.12 (page 204)
- Homework to be solved in Class!!!
- Problems 11, 26