Energy conservative (Kinetic Energy and Potential Energy)

[MechaMZ]

Homework Statement

May i know how do i compute an energy conservative equation for the scenario below, which i don't understand.

If a book is lifted up by a force in order to make it moving upward with a constant velocity.
For conservation of total energy, W_1-2 = U₁-U₂ = K₂ - K₁
If the object is moving in a constant velocity, isn't K₂ - K₁ = 0
However, U₁ is not equal to U₂.

It seems not correct obviously, hope someone could enlighten me.

I guess W_1-2 = U₁-U₂ = K₂ - K₁ is only valid for isolated system, which should be a system without any external force involves except gravitational force.

I'm very confuse now, hope someone could help me.

 

[staraet]

w=f.d
f=m.g
d=distance moved
earth does -ve work , you do equal +ve.


ok!

+ve change in potiential energy always needs external energy source.

so W=U2-U1=m.g.d

 

[PhanthomJay]

Homework Statement

May i know how do i compute an energy conservative equation for the scenario below, which i don't understand.

If a book is lifted up by a force in order to make it moving upward with a constant velocity.
For conservation of total energy, W_1-2 = U₁-U₂ = K₂ - K₁
If the object is moving in a constant velocity, isn't K₂ - K₁ = 0
However, U₁ is not equal to U₂.

It seems not correct obviously, hope someone could enlighten me.

The work done by non conservative forces (such as the lifting force on the book) is
W_{nc} = (U_2 - U_1) + (K_2 - K_1) or in its alternate form, W_{nc} = \Delta U + \Delta K

I guess W_1-2 = U₁-U₂ = K₂ - K₁ is only valid for isolated system, which should be a system without any external force involves except gravitational force.

I'm very confuse now, hope someone could help me.

For a system subject to conservative forces only (like gravity or spring forces), then (U_2 - U_1) + (K_2 -K_1) = 0, or \Delta U + \Delta K =0, and the work done by such conservative forces is W_c = U_1- U_2, or W_c = - (\Delta U) in its alternate form.

For a system subjct to both conservative and non conservative forces, then the total work done by those forces is W_T = (K_2- K_1), or W_T= \Delta K

staraet's answer for the work done by the lifting force is also correct.

This can get confusing for sure, but actually, I'm not exactly sure what your question is. Mechanical Energy (U + K) is NOT conserved when non-conservative forces that do work are acting.

 

[MechaMZ]

For a system subjct to both conservative and non conservative forces, then the total work done by those forces is W_T = (K_2- K_1), or W_T= \Delta K

why we no need to include the \Delta U? What if the object is in a falling motion with air resistive force acting on it.

Could i write it as: mg(\Delta h) - F_friction(\Delta h) = \Delta K + \Delta U

or i should write it as without taking the work done by gravitational force: - F_friction(\Delta h) = \Delta K +\Delta U

 

[staraet]

always think in terms of gain and loss

we some thing goes along only acting force , it gains k.e
so decreases its p.e

when ball falls, so it means, going along a force, so increases k.e (say after dt);
then if u find k.e is constant-- it means body has spent that k.e in either heat, vibrational,mass or something in that dt time.
then write equations. don't write before....

here friction is heat. so E spent in heat= mgh

 

[MechaMZ]

I think my problem is I don't understand between work done, kinetic energy, and potential energy.

 

[Redbelly98]

I suggest sticking with PhanthomJay's first equation:

The work done by non conservative forces (such as the lifting force on the book) is
W_{nc} = (U_2 - U_1) + (K_2 - K_1) or in its alternate form, W_{nc} = \Delta U + \Delta K.

As you realize for your example of lifting a book, K₂=K₁=0. Also, W_nc is the work W_1-2 done by lifting, so in that case

W_1-2 = U₂ - U₁​

If instead we have an isolated system with only conservative forces such as gravity, we have W_nc = 0 and conservation of mechanical energy applies. PhanthomJay's equation then becomes:

0 = U₂ - U₁ + K₂ - K₁​

this is often rewritten in the form

U₁ + K₁ = U₂ + K₂​

and is simply a statement that the total energy (potential + kinetic) does not change.