(Chap.6 p.139)

Work, Energy and Power

A.Prof. Hamid NEBDIDepartment of Physics

Faculty of Applied Science.

Umm Al-Qura University القرى أم معة جاFaculty of Applied Science كـلية العلوم التطبيقية General Physics 104 (For medical students) 104مدخل الفيزياء الطبية

1.Work• Suppose that an object is displaced a distance s, and that a force F acting on

the object has a constant component Fs along s.

Figure1 Figure 2

Then the work W done by the force F is defined as the product of

the force component Fs and the displacement :

W = Fs . s (1)If F is at an angle Ѳ to s, then Fs= F cos Ѳ, and the work can be

written as

W = F . s . cos Ѳ (2)The S.I unit of work is the joule (J). 1 joule = 1 N.m

Example 6.1:A 600 N force is applied by a man to a dresser that moves 2 m.

Find the work done if the force and displacement are (a) parallel;(b) at right angles;(c) oppositely directed (Fig.3); we may imagine that the dresser

is being slowed and brought to rest.

Figure 3

Solution 6.1:

(a) F and s are parallel, so cos Ѳ = cos 0⁰ = 1 and Eq. 2 gives:W = F . s . cos Ѳ = (600 N) (2m) (1) = 1200 J

The man does 1200 J of work on the dresser. Since F is parallel to s, Fs=F, and we obtain the same result using Eq. 1.

(b) F and s are perpendicular, so cos Ѳ = cos 90⁰ = 0 and Eq. 2 gives:W = F . s . cos Ѳ = (600 N) (2m) (0) = 0 J

No work is done when the force is at right angles to the displacement, since Fs = 0.

(c) F and s are opposite, so cos Ѳ = cos 180⁰ = -1 and Eq. 2 gives:W = F . s . cos Ѳ = (600 N) (2m) (-1) = -1200 J

In this case the work done by the force is negative, so the object is doing work on the man. Note that here F is opposite to s, so Fs = -F.

Example 6.2:A horse pulls a barge along a canal with a rope in which the

tension is 1000 N (Fig.4). The rope is at an angle of 10⁰ with the towpath and the direction of the barge.

(a) How much work is done by the horse in pulling the barge 100 m upstream at a constant velocity?

(b) What is the net force on the barge?

Figure 4

Solution 6.2:

(a)The work done by the constant force T in moving the barge a distance s is given by :

W = T . s . cos Ѳ = (1000 N) (100m) (cos 10⁰) =

(b)Since the barge moves at a constant velocity, the sum of all forces on it must be zero. There must be another force acting that is not shown in Fig. 4.

J410 x 85.9

Figure 4

• Remark:- If the force varies in magnitude or direction with respect to the displacement, the correct procedure is to consider the work done in a series of many small displacements. In each piece, we compute : ∆W = , where is the average force in this part of the motion. The sum of all these small terms then gives the net work done.- For computing this work, we can use the graphical method (Fig. 5); the area under the graph of Fs versus s for any motion is the work done.

sF s

cosFFs

Figure 5

sF

2. Kinetic Energy- The kinetic energy of an object is a measure of the work an object can do by virtue of its motion.

- The translational kinetic energy K of an object of mass m and velocity v is:

(3)- The work done on an object and its kinetic energy obey the

following fundamental principle (work-energy principal):The final kinetic energy (K) of an object is equal to itsinitial kinetic energy (K0) plus the total work (W) done on it by all the forces acting.

K = K0 + W (4)

(5)

2

2

1mvK

Figure 6

200 2

1mvK

Example 6.3:

Figure 7a

Solution 6.3:

3. Potential Energy- In general, potential energy is energy associated with the position or configuration of a mechanical system. In Fig. 8, a ball rises from initial height h0 to a height h.

The gravitational force mg is opposite in direction to the displacement s = h – h0, so the work done is negative:W(grav)= - mg (h-h0)

The change in potential energy is : ∆ U = U – U0

(6) The magnitude of ∆ U is defined to be equal to the magnitude of W(grav):

U – U0 = - W(grav) (7)This result for potential energy change involves a difference of two terms on each side, and suggests that we define the potential energy themselves at h and h0, respectively, by

U = mgh and U0 = mgh0 (8)

Figure 8

4. Total Energy- By the work-energy principal,

K = K0 + W(grav) = K0 - (U – U0)so we have the important result:

K + U = K0 + U0 (Wa = 0) (9)The notation Wa = 0 reminds us that the work done by applied forces is zero; only the gravitational forces is doing work here.- The sum of kinetic energy and the potential energy is called the total mechanical energy: E = K + U (10)Thus Eq. 9 means that when there is no work done by applied forces, the total mechanical energy is constant or conserved.- If applied forces also do a work, Eq. 9 must be generalized to include this work, Wa. Then we have:

K + U = K0 + U0 + Wa (11) or E = E0 + Wa (12)

The final mechanical energy E = K + U is equal to the initial mechanical energy E0 = K0 + U0 plus the work done by the applied forces.

Example 6.5:A woman skis from rest down a hill 20 m height(Fig. 9a).If friction is negligible, what is her speed at the

bottom of the slope?

Figure 9a

Solution 6.5:

Figure 9

-The work done by conservative forces can be handled conveniently

by the introduction of the potential energy concept.

- This is not true for frictional forces. There are treated as applied

forces: they give negative work.

- The work of these forces are converted in thermal energy:

dissipative forces.

5. Dissipative Forces

A woman skies from rest down a hill 20m high. If friction is not negligible and if her speed at the bottom of the slope is only 1ms-1, how much work is done by frictional forces if her mass is 50kg.

Solution 6.6:

Example 6.6:

Potential energy at the starts from rest:

Again, we choose the reference level : bottom of the slope so:

Using: E = E0 + Wa,

U=0

Kinetic energy at the top is: K0=0

U0=mgd

Kinetic energy at the bottom of the hill is:

Thus:

We can write: K+U=K0+U0+Wa

Hence:

6. Conservative Forces

- Any force that has the property that work it does is the same for all paths between two given points is said to be a conservative force.

- This property makes it meaningful to associate a potential energy with a position.

- Gravitational, electrical and spring forces are conservative forces.- Friction and many other forces are not conservative.

- In Figure 10, the work done by gravity is the same for the paths ABC and AC. Whenever the work done by a force is the same for all paths, the force is said to be conservative, and its effects can be included in the potential energy.

Figure 10

7. Power- When an amount of work ∆W is done in a time ∆t, the average power is defined as the average rate of doing work:

(10)

- The instantaneous power P is found by considering smaller and smaller time intervals, so : (11)

- The S.I. power unit is a joule per second, which is called a watt (W).- Energy is often sold by electrical utilities by the kilowatt hour (kW h).This is kilowatt of power for 1 hour. In terms of S.I. units,

- Another expression for the power is often useful. The work done by a force F acting through a small displacement ∆s in a short time ∆t is ∆W= Fs ∆s. Dividing by ∆t gives the power:

-Since ∆s /∆t is the velocity, the power is also given by:

t

WP

dt

dWP

JsWkWh 63 106.33600101

t

sFP s

vFP s

Example 6.14: (see p. 152)A 70-kg man runs up a flight of stairs 3m high in 2s. (a) How much work does he do against gravitational forces?(b) what is his average power output?

Solution 6.14: