Work done on free falling object as function of time

[pellikkan]

Hi there everyone,

Would like some checking of my work or comments.
Would like the work done on a falling mass M as a function of time.
To begin, the relation for work as function of distance..

w = Mgx

but would like as function of time so substitute x this way,

x = (1/2)gt²

implies work done as function of time would be

w = Mg(1/2)gt²

and finally for work as function of time;

w = (1/2)Mg²t²

Thanks in advance for any comments, verifications,
or kibbutzing in general!

 

[Doc Al]

Hi there everyone,

Would like some checking of my work or comments.
Would like the work done on a falling mass M as a function of time.
To begin, the relation for work as function of distance..

w = Mgx

but would like as function of time so substitute x this way,

x = (1/2)gt²

implies work done as function of time would be

w = Mg(1/2)gt²

and finally for work as function of time;

w = (1/2)Mg²t²

Thanks in advance for any comments, verifications,
or kibbutzing in general!

That's fine. That's the work done by gravity on a falling mass (starting from rest) after a time t.

 

[pellikkan]

Thanks there Doc for your kind and swift reply!

 

[sophiecentaur]

That's fine. That's the work done by gravity on a falling mass (starting from rest) after a time t.

A simpler way of stating it would be mgh, I think.

 

[Doc Al]

A simpler way of stating it would be mgh, I think.

That was the starting point, his first equation. But he wanted a function of time.

 

[sophiecentaur]

That'll teach me to read the whole thread.
Interestingly, the power increases with time.

 

[pellikkan]

Thanks SophieC for comments,
yep, that is interesting, ..if power
is defined as energy delivered within
a time period t, ie,

p = w/t

we'd get in that outlook,

p = (1/2)Mg²t,

which is linearly proportional to the time.

However, if I'm not mistaken, power I think might be supposed
to be more of an instantaneous concept, i.e.,
the small change in work done over a short time period;

p = Δw / Δt

in which case the problem might start to require
a bit of a calculus based treatment.

 

[Doc Al]

Thanks SophieC for comments,
yep, that is interesting, ..if power
is defined as energy delivered within
a time period t, ie,

p = w/t

we'd get in that outlook,

p = (1/2)Mg²t,

which is linearly proportional to the time.

However, if I'm not mistaken, power I think might be supposed
to be more of an instantaneous concept, i.e.,
the small change in work done over a short time period;

p = Δw / Δt

in which case the problem might start to require
a bit of a calculus based treatment.

Exactly right. What you've found is the average power over that time period. To find the instantaneous power, take the derivative: p = dw/dt.

Why don't you give it a shot?

 

[pellikkan]

well, i'll give that a shot,...

I 'd say that in a constant gravitational field, we know

w = mgx

But mg = const, so differentiating both sides ...

dw = mg dx

but since

x = (1/2)gt²

we differentiate each side to get;

dx = (1/2)g2t dt = gt dt

so that

dw = mg gt dt = mg²t dt

and dividing by dt on both sides,..

dw/dt = mg²t

but power is change in work within time period, so
the power is,.

P = mg²t

which is a weird result because the power delivered
to the system increases with time, whereas there is
still constant acceleration. It would be like if you had
a rocket with an engine that, even though it puts out a constant force,
the power it delivers keeps increasing.

We could check that result in another way,..
we can write that the work is the force times the distance,
written differentially with constant force,..

dw = F dx

and we could divide both sides by dt,..

dw/dt = F (dx/dt)

which reads...

P = Fv

where v is the velocity dx/dt.

which shows the same result, that the power
delivered increases with the velocity or time,
which would seem counter-intuitive a bit
since it appears you can't apply any power
to a stationary object,.. but we know that
to get your car moving the engine must be
running and engaged in delivering power and using gas.

Maybe it could be reckoned with by considering
your car or rocket tied to the ground with chains,
and if it can't move at all then you can apply
all the engine force you want but like it or not
you are not delivering any power to the car.

An interesting philosophical point is that if you're
dealing with an instantaneous new occurrence,
like a bomb going off you could employ the whole science of
statics instead of dynamics,... that is
sum of all force vectors = zero
ƩF = 0.
Since all velocities equal zero initially.
That is, you could introduce the forces of
the bomb into the building structure original blueprints, and using
the usual engineering science of statics you could
calculate all stresses throughout the building to see
where failure might occur, i.e., which stresses would be
enough for some failure,... i.e., acceleration of some part
of the building,.. in which case it would exit the science
of statics and enter the science of dynamics.

Sorry if I've rambled on a bit, it is late and just thinking out loud.

 

[Ryoko]

Wait a second ... you're trading potential energy for kinetic energy. The total energy remains the same, hence no work is performed.

 

[Emilyjoint]

Work = force x distance(basically)
Work is done on the falling mass by the force of gravity. This work done converts potential energy to kinetic energy (+ any energy due to air turbulence/friction)

 

[sophiecentaur]

Thanks SophieC for comments,
yep, that is interesting, ..if power
is defined as energy delivered within
a time period t, ie,

p = w/t

we'd get in that outlook,

p = (1/2)Mg²t,

which is linearly proportional to the time.

However, if I'm not mistaken, power I think might be supposed
to be more of an instantaneous concept, i.e.,
the small change in work done over a short time period;

p = Δw / Δt

in which case the problem might start to require
a bit of a calculus based treatment.

Difficult to avoid when you stray into Science!
There is no special significance to this speed - power thing except that we tend to think in terms of cars etc, which have a specific max power output and, hence, a top speed, defined by when engine power is balanced by 'speed times friction forces'. I often find it interesting just to turn things around and look at them slightly differently - producing counter-intuitive conclusions.
Of course, the process of falling under gravity has a limit too. When you hit the ground.