Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue...

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Chapter 6 Work and Energy

Transcript of Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue...

Page 1: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

Chapter 6

Work and Energy

Page 2: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

FsW =( )J joule 1 mN 1 =!

Work involves force and displacement.

Page 3: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

Page 4: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

( )sFW !cos=

1180cos

090cos

10cos

!=

=

=

o

o

o

More general definition of the work done on an object, W, by a constant force, F, through a displacement, s, with an angle, θ, between F and s:

Page 5: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

Example 1 Pulling a Suitcase-on-Wheels

Find the work done if the force is 45.0 N, the angle is 50.0 degrees, and the displacement is 75.0 m.

( ) ( )[ ]( )

J 2170

m 0.750.50cosN 0.45cos

=

== osFW !

Page 6: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

( ) FssFW == 0cos

( ) FssFW !== 180cos

A weightlifter doing positive andnegative work on the barbell:

Lifting phase --> positive work.

Lowering phase --> negative work.

Page 7: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

Example. Work done skateboarding. A skateboarder is coasting down aramp and there are three forces acting on her: her weight W (675 Nmagnitude) , a frictional force f (125 N magnitude) that opposes her motion,and a normal force FN (612 N magnitude). Determine the net work done bythe three forces when she coasts for a distance of 9.2 m.

W = (F cos θ)s , work done by each force

W = (675 cos 65o)(9.2) = 2620 J

W = (125 cos 180o)(9.2) = - 1150 J

W = (612 cos 90o)(9.2) = 0 J

The net work done by the three forces is: 2620 J - 1150 J +0 J = 1470 J

Page 8: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

Example 3 Accelerating a Crate

The truck is accelerating ata rate of +1.50 m/s2. The massof the crate is 120 kg and itdoes not slip. The magnitude ofthe displacement is 65 m.

What is the total work done on the crate by all of the forces acting on it?

Page 9: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

The angle between the displacementand the normal force is 90 degrees.

The angle between the displacement and the weight is also 90 degrees.

( ) 090cos == sFW

Thus, for FN and W,

Page 10: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.1 Work Done by a Constant Force

The angle between the displacementand the friction force is 0 degrees.

( )[ ]( ) J102.1m 650cosN180 4!==W

( )( ) N180sm5.1kg 120 2 === mafs

Page 11: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

Consider a constant net external force acting on an object.

The object is displaced a distance s, in the same direction asthe net force.

The work is simply

s

!F

W = (Σ F)s = (ma)s

Page 12: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

( ) ( ) 2212

2122

21

ofof mvmvvvmasmW !=!==

( )axvv of 222 +=

( ) ( )2221

of vvax !=

DEFINITION OF KINETIC ENERGY

The kinetic energy KE of and object with mass mand speed v is given by

221KE mv=

constant accelerationkinematics formula

Page 13: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

THE WORK-ENERGY THEOREM

When a net external force does work on an object, the kineticenergy of the object changes according to

2212

f21

of KEKE omvmvW !=!=

(also valid for curved paths and non-constant accelerations)

Page 14: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

Example 4 Deep Space 1

The mass of the space probe is 474 kg and its initial velocityis 275 m/s. If the 56.0 mN force acts on the probe through adisplacement of 2.42×109m, what is its final speed?

Page 15: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

2212

f21W omvmv !=

W = [(Σ F)cos θ]s

Page 16: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

( ) ( ) ( ) ( )( )2212

f2192- sm275kg 474kg 474m1042.20cosN105.60 !="" vo

( )[ ] 2212

f21cosF omvmvs !=" #

sm805=fv

[(Σ F)cos θ]s

Page 17: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

Example. A 58 kg skier is coasting down a 25o slope as shown. Near thetop of the slope, her speed is 3.6 m/s. She accelerates down the slopebecause of the gravitational force, even though a kinetic frictional forceof magnitude 71 N opposes her motion. Ignoring air resistance, determinethe speed at a point that is displaced 57 m downhill.

6.2 The Work-Energy Theorem and Kinetic Energy

Page 18: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

vo = 3.6 m/s s = 57 m 25o slope m = 58 kg fk = 71 N

Find vf

Page 19: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

kfmgF !=" o25sin

6.2 The Work-Energy Theorem and Kinetic Energy

2212

f21W omvmv !=

W = [(Σ F)cos θ]s

= (58)(9.8)(0.42) - 71 = 170 N

W = [(170)cos 0](57) = 9700 J

Solving the Work-Energy Theorem for vf:

vf = [(2W + mvo2)/m]1/2 = [(2(9700) + (58)(3.6)2)/(58)]1/2

= 19 m/s

Work-Energy Theorem

Page 20: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.2 The Work-Energy Theorem and Kinetic Energy

Conceptual Example 6 Work and Kinetic Energy

A satellite is moving about the earth in a circular orbit and an elliptical orbit. For these two orbits, determine whetherthe kinetic energy of the satellite changes during the motion.

Page 21: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.3 Gravitational Potential Energy

( )sFW !cos=

( )fo hhmgW !=gravity

Find the work done by gravity inaccelerating the basketballdownward from ho to hf.

Page 22: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.3 Gravitational Potential Energy

( )fo hhmgW !=gravity

We get the sameresult for Wgravityfor any path takenbetween ho and hf.

Page 23: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.3 Gravitational Potential Energy

Example 7 A Gymnast on a Trampoline

The gymnast leaves the trampoline at an initial height of 1.20 mand reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast?

Page 24: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.3 Gravitational Potential Energy

2212

f21W omvmv !=

( )fo hhmgW !=gravity

( ) 221

ofo mvhhmg !=!

( )foo hhgv !!= 2

( )( ) sm40.8m 80.4m 20.1sm80.92 2 =!!=ov

Page 25: Chapter 6The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured

6.3 Gravitational Potential Energy

fo mghmghW !=gravity

DEFINITION OF GRAVITATIONAL POTENTIAL ENERGY

The gravitational potential energy PE is the energy that anobject of mass m has by virtue of its position relative to thesurface of the earth. That position is measured by the heighth of the object relative to an arbitrary zero level:

mgh=PE

( )J joule 1 mN 1 =!