Query 142

course phy201

014. `query 14*********************************************

Question: `qset 3 intro prob sets

If you calculate the acceleration on a mass m which starts from rest under the influence of a constant net force Fnet and multiply by a

time interval `dt what do you get?

How far does the object travel during this time and what velocity does it attain?

What do you get when you multiply the net force by the displacement of the mass?

What kinetic energy does the object attain?

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Your solution:

we have

m

a

V0

fnet

and'dt

a * 'dt= 'dv or the change in velocity

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Given Solution:

`a The acceleration of the mass is a = F_net / m, so the velocity of the object changes by amount

`dv = a * `dt = F_net / m * `dt.

Since the initial velocity is zero, this will also be the final velocity:

vf = F_net / m * `dt.

From this and the fact that acceleration is constant (const. net force on const. mass implies const. acceleration), we conclude that

vAve = (v0 + vf) / 2 = (0 + (F_net / m) * `dt) / 2 = F_net * `dt / (2 m).

Multiplying this by the time interval `dt we have

`ds = vAve `dt = (F_net * `dt) / (2 m) * `dt = F_net `dt^2 / (2 m).

If we multiply this by F_net we obtain

F_net * `ds = F_net * F_net * `dt^2 / (2 m) = F_net^2 * `dt^2 / (2 m).

From our earlier result vf = F_net / m * `dt we see that

KE_f = 1/2 m vf^2 = 1/2 m ( F_net / m * `dt)^2 = F_net^2 * `dt^2 / (2 m).

Our final KE, when starting from rest, is therefore equal to the product F_net * `ds.

Since we started from rest, the final KE of the mass on this interval is equal to the change in KE on the interval.

We call F_net * `ds the work done by the net force. Our result therefore confirms the work-kinetic energy theorem:

`dW_net = `dKE.

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Self-critique (if necessary):

I understand all the formulas I think.

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Self-critique rating #$&*

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Question: `q Define the relationship between the work done by a system against nonconservative forces, the work done against

conservative forces and the change in the KE of the system. How does PE come into this relationship?

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Your solution:

the work done BY a system is positive as nonconservative forces do work on a system are negative. Work done against conservative forces

is negative as conservative forces are doing positive work By a system. The change in KE is negative if work done is negative or

positive if work done is positve and PE is opposing KE, so if Ke is positve PE will be negative.

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Given Solution:

`a** The system does positive work at the expense of its kinetic and/or potential energy.

The work done by the system against all forces is `dW_net_BY.

`dW_net_BY is equal and opposite to `dW_net_ON, which is in turn equal to `dKE, the change in the kinetic energy of the system.

We conclude that `dW_net_BY = - `dKE. The change in KE is equal and opposite to the work done by the system against the net force

acting on it.

To consider the role of PE, we first review our formulation in terms of the work done ON the system:

`dW_net_ON = `dKE.

The work `dW_net_ON is the sum of the work done on the system by conservative and nonconservative forces:

`dW_net_ON = `dW_cons_ON + `dW_NC_ON

and `dW_cons_ON is equal and opposite to `dPE, the change in the system's PE.

Thus `dW_net_ON = `dW_NC_ON - `dPE so that `dW_net_ON = `dW_cons_ON + `dW_NC_ON becomes

`dW_NC ON - `dPE = `dKE so that

`dW_NC_ON = `dPE + `dKE.

Since `dW_NC_BY = - `dW_NC_ON, we see that

-`dW_NC_BY = `dPE + `dKE so that

`dW_NC_BY + `dPE + `dKE = 0.

Intuitively, if the system does positive work against nonconservative forces, `dPE + `dKE must be negative, so the total mechanical

energy PE + KE of the system decreases. (Similarly, if the system does negative work against nonconservative forces that means

nonconservative forces are doing positive work on it, and its total mechanical will increase).

As usual, you should think back to the most basic examples in order to understand all these confusing symbols and subscripts (e.g., if I

lift a mass, which you know intuitively increases its gravitational potential energy, I do positive work ON the system consisting of the

mass, the conservative force of gravity acts in the direction opposite motion thereby doing negative work ON the system, and the work

done BY the system against gravity (being equal and opposite to the work done ON the system by gravity) is therefore positive).

The equation -`dW_NC_BY = `dPE + `dKE isolates the work done by the system against nonconservative forces from the work it does against

conservative forces, the latter giving rise to the term `dPE.

If the system does positive work against conservative forces (e.g., gravity), then its PE increases.

If the system does positive work against nonconservative forces (e.g., friction) then `dPE + `dKE is negative: PE might increase or

decrease, KE might increase or decrease, but in any even the total PE + KE must decrease. The work done against a nonconservative force

is done at the expense of at least one, and maybe both, the PE and KE of the system. (In terms of gravitational forces, the system gets

lower or slows down, and maybe both, in order to do the work).

If nonconservative forces do positive work on the system, then the system does negative work against those forces, and `dW_NC_ON is

negative. Thus -`dW_NC_ON is positive, and `dPE + `dKE is positive. Positive work done on the system increases one or both the PE and

the KE, with a net increase in the total of the two. (In terms of gravitational forces, the work done on system causes it to get higher

or speed up, and maybe both.)

STUDENT RESPONSE WITH INSTRUCTOR COMMENTARY: The work done by a system against nonconservative forces is the work done to overcome

friction in a system- which means energy is dissipated in the form of thermal energy into the 'atmosphere.'

Good. Friction is a nonconservative force.

However there are other nonconservative forces--e.g., you could be exerting a force on the system using your muscles, and that force

could be helping or hindering the system. A rocket engine would also be exerting a nonconservative force, as would just about any

engine. These forces would be nonconservative since once the work is done it can't be recovered.

STUDENT RESPONSE WITH INSTRUCTOR COMMENTS: The work done by a system against conservative forces is like the work to overcome the mass

being pulled by gravity.

INSTRUCTOR COMMENT: that is one example; another might be work to compress a spring

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Self-critique (if necessary):

????? ok, alot to absorb but I think I am getting there. So KE is equal to work done ON the systme not BY the system...this is a little

confusing. I was thinking that when we calculate the dw= fnet * 'ds we were caculated the work done BY the system not on the

system.?????

I've attempted to address your question below:

To be very specific

is the work done by the net force acting ON the system, where

F_net_ON is the net force acting on the system.


The work-kinetic energy theorem states that

If positive work is done on a system, it speeds up. If

negative work is done on the system, it slows down.


From the point of view of the system, if positive work is done by the system

then the system has to 'use up' some of its kinetic energy to do the work, so it

slows.


Positive work done BY the system constitutes negative work being done ON the

system.


If part of the net force is conservative, then `dW_net_ON can be split

into `dW_net_ON_cons and `dW_net_ON_noncons.

The quantity `dPE, the change in PE, is defined to

be equal and opposite to `dW_net_ON_cons. That is,

It follows that `dW_net_ON = `dW_net_ON_noncons - `dPE,

so that the work-kinetic energy theorem can be rewritten as

This is commonly rearranged to the form