gravity 2. Work done by a variable force, Exercises of Physics

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Work

Sections 7.6 – 7.

Work done by a constant force: gravity

Work done by a variable force: a spring

Work done by a variable force: general case

Work-kinetic energy theorem

The theorem says that the change in kinetic energy of a

particle is the net work done on the particle.

It holds for both positive and negative work: If the net workdone on a particle is positive, then the particle’s kineticenergy increases by the amount of the work, and theconverse is also true.

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Solution = E

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Lecture 11 – Work

Solution

K

f

= ½

m

v

i

2

• net work done by forces on box

= ½(2 kg)(3 m/s)

2

W

net

= 9 J +

W

net

(a)

K

f

= 9 J + [(10 N)cos

o

• (2 N)](4 m) = 52 J

(b)

K

f

=

9 J + [(10 N)cos

o

• (2 N)](4 m) = 36 J

W = K

f

- K

i

(c)

K

f

= 9 J + (10 N)(4 m) = 49 J

(d)

K

f

= 9 J + [(10 N)sin

o

• (2 N) sin

o

](4 m) = 25 J

Answer:

a, c, b, d

Work Done by the Gravitational Force Work Done by the Gravitational Force

(

a) An applied force

lifts

an object

The object’s displacement makes anangle

φ

=180° with the gravitational

force on the objectThe applied force does positive workon the object.

(

b) An applied force

lowers

an object.

The displacement of the object makesan angle with the gravitational force.The applied force does negative workon the object

W

g

= mgd

cos

o

= -

mgd

W

g

= mgd

cos

o

=

mgd

W

a

= mgd

cos

o

=

mgd

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The work done by gravity during the descent of a projectile is:

A) positive B) negativeC) zeroD) depends for its sign on the direction of the

y

axis

E) depends for its sign on the direction of both the

x

and

y

axes

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Lecture 11 – Work and Power

Work Done by a Spring

k

is called the

spring constant

The spring constant measures the stiffness of thespring

k

has units of N/m.

Hooke’s Law,

F

(

x

)

= -kx,

is a

variable force

The force depends on displacement in a linearmanner

07/10/

Lecture 11 – Work and Power

Work Done by a Spring

Since the force applied by the spring is not constant,how do we find the work?

We know

W = F x

is true when

F

is constant, so we do

the following:

• Divide interval into small segments

Small enough so in

∆

x

,

F

is constant!

• Find the work in each segment– Add them up!

W

Σ(

F

i

x

Magnitude of Fonly shown here.

07/10/

Lecture 11 – Work and Power

Work Done by a Spring

What is the more sophisticated method for adding under acurve?

Integration!

Integrating the force from the initial to final position of the massgives us

2

2

f

i

x

x

x

x

kx

kx

dx

kx

Fdx

W

f i f i

Work done by a spring force:

Initialtermfirst!

Initialtermfirst!

Work Done by a Spring Force Work Done by a Spring Force

Hooke’s Law:

To a good approximation for many springs, the force from a spring is

proportional to the displacement of the free end from its position when the spring is in therelaxed state. The

spring force is given by

The minus sign indicates that the direction of the spring force is always opposite the directionof the displacement of the spring’s free end.The constant

k is called the

spring constant (or force constant) and is a measure

of the

stiffness of the spring.The net work

W

s

done by a spring, when it has a distortion from x

i

to x

f

, is:

kx

F

s

−

=

Work W

s

is positive if the block ends up closer to the relaxed position (x =0) than it was

initially. It is negative if the block ends up farther away from x =0. It is zero if the block ends

up at the same distance from x= 0.

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Work Done by a Force Applied to the Block Work Done by a Force Applied to the Block

Apply a force, Apply a force,

F

F

aa

, to a block which has a spring connected on , to a block which has a spring connected on

one side. The force results in compression of the spring. one side. The force results in compression of the spring.

As you do work As you do work

W

W

a a

on the block, the spring does work, on the block, the spring does work,

W

W

s s

, on , on

the block the block

Work Work

KE theorem: KE theorem:

K

K

f f

K

K

i i

W

W

a a

+ W

+ W

ss

If the block is stationary before and after the displacement the If the block is stationary before and after the displacement the

n n

K

K

f f

K

K

ii

= 0, so = 0, so

W

W

a a

WW

ss

What does this mean physically? What does this mean physically?

Lecture 11, Work and PowerLecture 11, Work and Power

07/10/ 07/10/

Work Done by an Applied Force Work Done by an Applied Force

Work applied to block is equal and opposite to the work done by Work applied to block is equal and opposite to the work done by

the spring the spring

If the block is If the block is

not not

stationary after the displacement, is this stationary after the displacement, is this

statement true? statement true?

NO!, since NO!, since

W

W

s s

K

K

f f

K

K

i i

WW

aa

the work done by the spring would the work done by the spring would

also depend on also depend on

K

K

Lecture 11, Work and PowerLecture 11, Work and Power

07/10/ 07/10/

Sign Convention: work done by spring force Sign Convention: work done by spring force

W W

=

=

½

½

k

k

(

(

x x

i i

2 2

x x

f f

2 2

) )

W

W

= 0 = 0

is true when |

is true when |

x

x

i i

x

x

f f

|, and the block ends up at the

|, and the block ends up at the

same distance from the equilibrium position it

same distance from the equilibrium position it

started at

started at

means no energy is transferred

means no energy is transferred

Lecture 11, Work and PowerLecture 11, Work and Power

07/10/ 07/10/

Sign Convention: work done by spring force Sign Convention: work done by spring force

W W

=

=

½

½

k

k

(

(

x x

i i

2 2

x x

f f

2 2

) )

What is the physical meaning of the absolute value

What is the physical meaning of the absolute value

signs, |

signs, |

x

x

|, for each of these situations?

|, for each of these situations?

Consider Consider

WW

= 0. The block ends up at the same distance from = 0. The block ends up at the same distance from

the equilibrium point, but it could be physically on the the equilibrium point, but it could be physically on the

opposite side from whence it started opposite side from whence it started

Since work depends on the square of position, the distance Since work depends on the square of position, the distance

from the equilibrium point can be determined, but not the from the equilibrium point can be determined, but not the

displacement vector! displacement vector!

Lecture 11, Work and PowerLecture 11, Work and Power