# 7.5 Nonconservative forces  (Page 2/5)

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${W}_{\text{net}}={W}_{\text{nc}}+{W}_{\text{c}},$

so that

${W}_{\text{nc}}+{W}_{c}=\text{Δ}\text{KE},$

where ${W}_{\text{nc}}$ is the total work done by all nonconservative forces and ${W}_{\text{c}}$ is the total work done by all conservative forces.

Consider [link] , in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational potential energy, so that ${W}_{\text{c}}=-\text{Δ}\text{PE}$ . Substituting this equation into the previous one and solving for ${W}_{\text{nc}}$ gives

${W}_{\text{nc}}=\text{Δ}\text{KE}+\text{Δ}\text{PE.}$

This equation means that the total mechanical energy $\left(\text{KE + PE}\right)$ changes by exactly the amount of work done by nonconservative forces. In [link] , this is the work done by the person minus the work done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy.

We rearrange ${W}_{\text{nc}}=\text{Δ}\text{KE}+\text{Δ}\text{PE}$ to obtain

${\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}+{W}_{\text{nc}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}\text{.}$

This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If ${W}_{\text{nc}}$ is positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in [link] . If ${W}_{\text{nc}}$ is negative, then mechanical energy is decreased, such as when the rock hits the ground in [link] (b). If ${W}_{\text{nc}}$ is zero, then mechanical energy is conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a constant energy.

## Applying energy conservation with nonconservative forces

When no change in potential energy occurs, applying ${\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}+{W}_{\text{nc}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}$ amounts to applying the work-energy theorem by setting the change in kinetic energy to be equal to the net work done on the system, which in the most general case includes both conservative and nonconservative forces. But when seeking instead to find a change in total mechanical energy in situations that involve changes in both potential and kinetic energy, the previous equation ${\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}+{W}_{\text{nc}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}$ says that you can start by finding the change in mechanical energy that would have resulted from just the conservative forces, including the potential energy changes, and add to it the work done, with the proper sign, by any nonconservative forces involved.

## Calculating distance traveled: how far a baseball player slides

Consider the situation shown in [link] , where a baseball player slides to a stop on level ground. Using energy considerations, calculate the distance the 65.0-kg baseball player slides, given that his initial speed is 6.00 m/s and the force of friction against him is a constant 450 N.

Strategy

Friction stops the player by converting his kinetic energy into other forms, including thermal energy. In terms of the work-energy theorem, the work done by friction, which is negative, is added to the initial kinetic energy to reduce it to zero. The work done by friction is negative, because $\mathbf{\text{f}}$ is in the opposite direction of the motion (that is, $\theta =\text{180º}$ , and so $\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =-1$ ). Thus ${W}_{\text{nc}}=-\text{fd}$ . The equation simplifies to

$\frac{1}{2}{{\text{mv}}_{i}}^{2}-\text{fd}=0$

or

$\text{fd}=\frac{1}{2}{{\text{mv}}_{i}}^{2}\text{.}$

This equation can now be solved for the distance $d$ .

Solution

Solving the previous equation for $d$ and substituting known values yields

$\begin{array}{lll}d& =& \frac{{{\text{mv}}_{i}}^{2}}{2f}\\ & =& \frac{\left(\text{65.0 kg}\right)\left(6\text{.}\text{00 m/s}{\right)}^{2}}{\left(2\right)\left(\text{450 N}\right)}\\ & =& \text{2.60 m.}\end{array}$

Discussion

The most important point of this example is that the amount of nonconservative work equals the change in mechanical energy. For example, you must work harder to stop a truck, with its large mechanical energy, than to stop a mosquito.

When using the Conservation of Energy equation, do we substitute the energy as a negative quantity when the energies on a single object are exerting forces opposite to one another?
Ex. On an inclined plane, gravitational potential energy, friction energy/work and spring potential energy. (Let's say that the spring is keeping the box from sliding down the slope.) How do we use this in the equation? I'm so confused
Jennifer
Oh! And if there's kinetic energy that is exerting a force opposite to the spring, what do we do?
Jennifer
why is it dat when using double pan balance the known and unknown mass are the same
discuss the uses of energy in the following sectors of economy security and education
amajuoyi
is there more then 4 dimensions
hii
princy
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Miguel
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Brown
Brown. what kind of help
Jeff
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Jeff
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ayesha
sticking to the basics will take you farther than overwhelming yourself with more than you need to physics is simple keep it simple
Jeff
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Jeff
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Brown
read Stephan hawkings a brief history of time
ayesha
ayesha
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ayesha
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Brown
ayesha
ayesha
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Varun
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it what?
Luke
in physics
ayesha
the basic formula is uncertainty in momentum multiplied buy uncertainty In position is greater than or equal to 4×pi/2. same formula for energy and time
Luke
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ayesha
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uma
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Suthar
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farooq
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Iram
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what is the dimension of strain
Is there a formula for time of free fall given that the body has initial velocity? In other words, formula for time that takes a downward-shot projectile to hit the ground. Thanks!
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Agboro
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Chandan
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Sahim
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Jeff
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Priscilla
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Bile
Hy
Kulsum
What is unit of watt?
Kulsum
watt is the unit of power
Rahul
p=f.v
Rahul
watt can also be expressed as Nm/s
Rahul
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Maxamed
SI unit of mass is Kg(kilogram).
Robel
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Maxamed
Formula for for the falling body with initial velocity is:v^2=v(initial)^2+2*g*h
Mateo
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Maxamed
we can't do this calculation without knowing the height of the initial position of the particle
Chathu
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Imoreh
2 forces whose resultant is 100N, are at right angle to each other .if one of them makes an angle of 30 degree with the resultant determine it's magnitude
50 N... (50 *1.732)N
Sahim
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Sahim
50 N...(50 *1.732)N
Ibrahim
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usiomon
what is the value of f1 and f2
Syed
what is the value of force 1 and force 2.
Syed
.
Is earth is an inertial frame?
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system
Sahim
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Irungu
Most welcome
Sahim
Hey.. I've a question.
Is earth inertia frame?
Sahim
only the center
Shii
What is an abucus?
Irungu
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
prakash
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
so solve it
Ebenezer
Ebenezer
two bodies x and y start from rest and move with uniform acceleration of a and 4a respectively. if the bodies cover the same distance in terms of tx and ty what is the ratio of tx to ty
what is cesium atoms?
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
what is difference between entropy and heat capacity
Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
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sir how to understanding clearly
Vinodhini
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revolutionary
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Vinodhini
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Vinodhini
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