In general, forces may vary in magnitude and direction at points in space, and paths between two points may be curved. The infinitesimal work done by a variable force can be expressed in terms of the components of the force and the displacement along the path,
$dW={F}_{x}dx+{F}_{y}dy+{F}_{z}dz.$
Here, the components of the force are functions of position along the path, and the displacements depend on the equations of the path. (Although we chose to illustrate
dW in Cartesian coordinates, other coordinates are better suited to some situations.)
[link] defines the total work as a line integral, or the limit of a sum of infinitesimal amounts of work. The physical concept of work is straightforward: you calculate the work for tiny displacements and add them up. Sometimes the mathematics can seem complicated, but the following example demonstrates how cleanly they can operate.
Work done by a variable force over a curved path
An object moves along a parabolic path
$y=(0.5\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}){x}^{2}$ from the origin
$A=(0,0)$ to the point
$B=(2\phantom{\rule{0.2em}{0ex}}\text{m,}\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}\text{m})$ under the action of a force
$\overrightarrow{F}=(5\phantom{\rule{0.2em}{0ex}}\text{N/m})y\widehat{i}+(10\phantom{\rule{0.2em}{0ex}}\text{N/m})x\widehat{j}$ (
[link] ). Calculate the work done.
Strategy
The components of the force are given functions of
x and
y . We can use the equation of the path to express
y and
dy in terms of
x and
dx ; namely,
This integral was not hard to do. You can follow the same steps, as in this example, to calculate line integrals representing work for more complicated forces and paths. In this example, everything was given in terms of
x - and
y -components, which are easiest to use in evaluating the work in this case. In other situations, magnitudes and angles might be easier.
Check Your Understanding Find the work done by the same force in
[link] over a cubic path,
$y=(0.25\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\text{\u22122}}){x}^{3}$ , between the same points
$A=(0,0)$ and
$B=(2\phantom{\rule{0.2em}{0ex}}\text{m,}\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}\text{m})\text{.}$
You saw in
[link] that to evaluate a line integral, you could reduce it to an integral over a single variable or parameter. Usually, there are several ways to do this, which may be more or less convenient, depending on the particular case. In
[link] , we reduced the line integral to an integral over
x , but we could equally well have chosen to reduce everything to a function of
y . We didn’t do that because the functions in
y involve the square root and fractional exponents, which may be less familiar, but for illustrative purposes, we do this now. Solving for
x and
dx , in terms of
y , along the parabolic path, we get
In Example, we calculated the final speed of a roller coaster that descended 20 m in height and had an
initial speed of 5 m/s downhill. Suppose the roller coaster had had an initial speed of 5 m/s uphill instead,
and it coasted uphill, stopped, and then rolled back down to a final point 20 m bel
A steel lift column in a service station is 4 meter long and .2 meter in diameter. Young's modulus for steel is 20 X 1010N/m2. By how much does the column shrink when a 5000- kg truck is on it?
no, it just means that a brick isn't gonna move unless something makes it move. if in the air, moves down because of gravity. if on floor, doesn't move unless something has it move, like a hand pushing the brick. first law is that an object will stay at rest or motion unless another force acts upon
Grant
yeah but once gravity has already been exerted .. i am saying that it need not be constantly exerted now according to newtons first law
Dharmee
gravity is constantly being exerted. gravity is the force of attractiveness between two objects. you and another person exert a force on each other but the reason you two don't come together is because earth's effect on both of you is much greater
Grant
maybe the reason we dont come together is our inertia only and not gravity
Dharmee
this is the definition of inertia: a property of matter by which it continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force.
Grant
the earth has a much higher affect on us force wise that me and you together on each other, that's why we don't attract, relatively speaking of course
Grant
quite clear explanation but i just want my mind to be open to any theory at all .. its possible that maybe gravity does not exist at all or even the opposite can be true .. i dont want a fixed state of mind thats all
Dharmee
why wouldn't gravity exist? gravity is just the attractive force between two objects, at least to my understanding.
Grant
earth moves in a circular motion so yes it does need a constant force for a circular motion but incase of objects on earth i feel maybe there is no force of attraction towards the centre and its our inertia forcing us to stay at a point as once gravity had acted on the object
Dharmee
why should it exist .. i mean its all an assumption and the evidences are empirical
Dharmee
We have equations to prove it and lies of evidence to support. we orbit because we have a velocity and the sun is pulling us. Gravity is a law, we know it exists.
Grant
yeah sure there are equations but they are based on observations and assumptions
Dharmee
g is obtained by a simple pendulum experiment ...
Dharmee
gravity is tested by dropping a rock...
Grant
and also there were so many newtonian laws proved wrong by einstein . jus saying that its a law doesnt mean it cant be wrong
Dharmee
pendulum is good for showing energy transfer, here is an article on the detection of gravitational waves: ***ligo.org/detections.php
Grant
yeah but g is calculated by pendulum oscillations ..
Dharmee
thats what .. einstein s fabric model explains that force of attraction by sun on earth but i am talking about force of attraction by earth on objects on earth
Dharmee
no... this is how gravity is calculated:F = G*((m sub 1*m sub 2)/r^2)
Grant
gravitational constant is obtained EXPERIMENTALLY
Dharmee
the G part
Dharmee
Calculate the time of one oscillation or the period (T) by dividing the total time by the number of oscillations you counted. Use your calculated (T) along with the exact length of the pendulum (L) in the above formula to find "g." This is your measured value for "g."
Dharmee
G is the universal gravitational constant. F is the gravity
Grant
search up the gravity equation
Grant
yeahh G is obtained experimentally
Dharmee
sure yes
Grant
thats what .. after all its EXPERIMENTALLY calculated so its empirical
Dharmee
yes... so where do we disagree?
Grant
its empirical whixh means it can be proved wrong
Dharmee
so cant just say why wouldnt gravity exists
Dharmee
the constant, sure but extremely unlikely it is wrong. gravity however exists, there are equations and loads of support surrounding the concept. unfortunately I don't have a high enough background in physics but have this discussion with a physicist
Grant
can u suggest a platform where i can?
Dharmee
stack overflow
Grant
stack exchange, physics section***
Grant
its an app?
Dharmee
there is! it is also a website as well
Grant
okayy
Dharmee
nice talking to you
Dharmee
***physics.stackexchange.com/
Grant
likewise :)
Grant
What is the percentage by massof oxygen in Al2(so4)3
A spring with 50g mass suspended from it,has its length extended by 7.8cm
1.1 determine the spring constant?
1.2 it is observed that the length of the spring decreases by 4.7cm,from its original length, when a toy is place on top of it. what is the mass of the toy?
solution
mass = 50g= 0.05kg
force= 50 x 10= 500N
extension= 7.8cm = 0.078m
using the formula
Force= Ke
K = force/extension
500/.078
= 6410.25N/m
Sampson
1.2
Decrease in length= -4.7cm =-0.047m
mass=?
acceleration due to gravity= 10
force = K x e
force= mass x acceleration
m x a = K x e
mass = K x e/acceleration
= 6410.25 x 0.047/10
= 30.13kg
Sampson
1.1 6.28Nm-¹
Anita
1.2 0.03kg or 30g
Anita
I used g=9.8ms-²
Anita
you should explain how yoy got the answer Anita
Grant
ok
Anita
with the fomular F=mg I got the value for force because now the force acting on the spring is the weight of the object and also you have to convert from grams to kilograms and cm to meter
Anita
so the spring constant K=F/e
where
F is force and e is extension
Anita
In this first example why didn't we use P=P° + ¶hg
where ¶ is density
same acceleration for all particles because all prticles will be moving with same angular velocity.so at any time interval u find same acceleration of all the prticles
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass
that is f=ma
David
The rate of change of momentum of a body is directly proportional to the force exerted on that body.
Rani
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?