52
votes
Accepted
If work is a scalar measurement, why do we sometimes represent it as the product of force (a vector) and distance (scalar)?
Work is the dot product of a vector force and a vector displacement, hence a scalar.
Knowing just the scalar distance isn’t enough to calculate work. That distance might be in the same direction as ...
- 14.4k
37
votes
Accepted
If application of force does not result in spatial movement, has work been done?
No. No work would be done in this case, at least not at the macroscopic level. Work is the product of force and displacement in the direction of the force and in this case there is no displacement.
...
- 8,640
30
votes
If work is a scalar measurement, why do we sometimes represent it as the product of force (a vector) and distance (scalar)?
The general definition of work is
$$W=\int\mathbf F\cdot\text d\mathbf x$$
Which essentially says, "Add up all of the dot products between the vector force $\mathbf F$ and the vector displacement $\...
- 55.6k
24
votes
Why is work done not equal to force times time?
Why is work done not equal to force times time?
You have definitions backwards. It's not like we said "Ah yes, 'work' is important, what should its definition be?" The reason work is ...
- 55.6k
21
votes
Accepted
Why do we equate an indefinite integral to a specific value?
In physics we frequently leave off the limits of the integral when the limits can be figured out from the context. So, in the first case, the actual relation is:
$$x(t) = x(0) + \int_0^t \dot{x} \...
- 22.2k
16
votes
Why is small work done always taken as $dW=F \cdot dx$ and not $dW=x \cdot dF$?
Very nice question!
You can see this from the second Newton's law:
$$m\ddot{\mathbf{x}} = \mathbf{F}(\mathbf{x})$$
Now I would like to integrate this equation of motion with respect to time, to ...
- 4,792
16
votes
What does the 'displacement' refer to in the definition of work?
The distance referred to in the definition of work is, specifically, the distance that the object moves while the force is being applied. This is because the actual definition of work is a line ...
- 35.2k
12
votes
Why do we equate an indefinite integral to a specific value?
You are being confused by the shortcut that people took who wrote that expression. They mean for the integral to be taken between definite limits.
It would be more proper to say
$$x(t) = x(0) + \...
- 118k
12
votes
Why is small work done always taken as $dW=F \cdot dx$ and not $dW=x \cdot dF$?
The answer to your question depends on how we define work.
Definition, A force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force.
...
- 4,877
12
votes
Why is small work done always taken as $dW=F \cdot dx$ and not $dW=x \cdot dF$?
There are already several good answers. In this answer, we will just highlight a geometric argument.
On one hand, work (within Newtonian mechanics)
$$\mathrm{d}W=\vec{F}\cdot \mathrm{d}\vec{r}$$ is a ...
- 193k
12
votes
Accepted
If displacement is 0, does that mean initial velocity equals final velocity?
As @AccidentalTaylorExpansion and @David White discuss in their answers, your relationship is only valid for constant linear acceleration. Your situation is rotational, not linear motion. Also, ...
- 9,313
9
votes
Why is work done not equal to force times time?
You cannot set up a machine that "uses 1 Joules/s to exert a force of 1N on any object", precisely because it would lead to the sort of contradictory statements about work and energy spent ...
- 120k
9
votes
If the displacement of an object is not differentiable at some point, say $x(t)=t\sin(1/t)$ at $t=0$, how is its instant $v$ defined?
If $x(t)$ isn't differentiable at some $t_0$, then $v(t_0)\equiv x'(t_0)$ isn't defined. That's what it means for a function not to be differentiable.
If you argue that the instantaneous velocity of ...
- 66k
9
votes
Does work done by a non-conservative force involve distance rather than displacement?
Both conservative and nonconservative forces do work as the path integral $\int _L \vec F \cdot d\vec s$.
If force and path are antiparallel (as for friction*) and force is constant in magnitude along ...
- 12.6k
8
votes
Why the name 'displacement' operator?
A coherent state is characterized by a complex number $\alpha \in \mathbb C$.
Applying the displacement operator $D(\beta)$ to $|\alpha\rangle$ translates $\alpha$ in the complex plane by $\beta$, in ...
- 7,023
8
votes
Does Euler number $e$ have a role in kinematics?
A simple example is just an object starting at rest falling with a drag force proportional to the velocity of the object, $F_D=-bv$. Then the acceleration is given by
$$a=\frac{dv}{dt}=g-\frac bmv$$
...
- 55.6k
8
votes
Why is work defined as $W=Fd$?
The first thing you need to understand: you are applying the creation of physics definitions backwards. You are asking, "why isn't work given by this equation?", but this question doesn't make sense ...
- 55.6k
7
votes
Accepted
Why is work scalar and the dot product of force and displacement?
Well, say that you are looking at a man lifting boxes. Each box weighs 10kg. At first you look at him standing, but since just looking at him made you tired, you decide to lie down. Now, from your ...
- 5,099
7
votes
Paradox in the Kinematic SUVAT Equations of Motion
There is no paradox, because your 5 equations are not independent.
Fill in (1) in (3) to obtain (2)
(2)+(3) = 2*(4)
(5) is left as an exercise to the reader.
You also see this in your example. ...
- 5,050
6
votes
Are distance and displacement always frame independent?
The following discussion disregards effects of special relativity and discusses purely Galilean transformations.
If two frames are at rest with respect to each other and oriented parallel to one ...
- 1,796
6
votes
Accepted
Does Euler number $e$ have a role in kinematics?
Of course, $e$ is ubiquitous in kinematics. For example, consider a repulsive force proportional to $x$,
$$F = kx.$$
Then the acceleration is
$$a = \frac{k}{m} x = \omega^2 x, \quad \omega = \sqrt{\...
- 99.9k
6
votes
Accepted
What is the difference between position, displacement, and distance traveled?
Position is a single point. Usually in space we indicate positions with coordinates like $(x,y,z)$ in Cartesian coordinates, $(r,\phi,\theta)$ in spherical coordinates, etc. We can also define the ...
- 55.6k
6
votes
What is the difference between position, displacement, and distance traveled?
$\color{blue}{\text{Position}}$:
$$\color{blue}{\vec p(t)}$$
$\color{red}{\text{Displacement}}$†:
$$\color{red}{\vec p(t_2) - \vec p(t_1)}$$
$\color{green}{\text{Distance Traveled}}$:
$$\color{green}{\...
- 1,844
6
votes
Definition of velocity in the context of affine space
In affine spaces the derivative of a curve $P=P(t)$ is a vector since couples of points uniquely define vectors:
$$P(t + h)-P(t)$$
as a consequence of the affine structure.
As it is a well-defined ...
- 67.9k
5
votes
Why do we need displacement?
If you drive along the road on the side of a mountain) there are (at least) two kinds of forces on your car: friction and gravity.
If you drive to the top of the mountain and back, the net work done ...
- 118k
5
votes
If application of force does not result in spatial movement, has work been done?
Their lesson today says that "work" is done only when a change in position is accomplished by application of force.
The key word here is change. If an object is not moving, then no work is being done ...
- 55.6k
5
votes
If displacement is 0, does that mean initial velocity equals final velocity?
The kinematic equations only apply to situations where the acceleration is constant. In your example the guy on is bike is accelerating at different rates so the kinematic equations do not apply. That'...
5
votes
Is velocity real?
Displacement which you understand, and time which you understand, form the ratio velocity which is displacement divided by time.
Consider an object that travels $100 m$ in ten seconds. That means its ...
- 28.7k
5
votes
Why is position proportional to time squared?
When an object initially at rest has uniform acceleration $a$ for time $t$, it reaches speed $at$. Its average speed up to time $t$ is $(0+at)/2=\frac12at$. Therefore, the distance travelled is $\...
- 24.6k
5
votes
Springs stacked on each other in series with a mass on top, is the deformation the same?
Imagine one single coil of wire making a spring of length L and constant k. You can cut it in half to get two springs, each with length L/2 and constant k. If you stack the two springs, you have an ...
- 8,902
Only top scored, non community-wiki answers of a minimum length are eligible