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Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?

I think its "φ/ω" http://www.intmath.com/trigonometric-graphs/3-graphs-sin-cos-phase-shift.php

There are two waves y1=a. sin(ω.t) & y2=a. sin(ω.t + φ) My question is what is the phase difference of these two waves? Is it "φ" or "φ/ω"?

I did not get my answer from that. I need the derivation

I understand the derivation of gravitational potential energy when an object is moved away from Earth but when I try to derive the formula for gpe by considering a test mass moving from infinity to r then I end up with a positive gravitational potential energy. Because integrating F.dr from...

Why did you put the magnitude of force, shouldn't it contain minus sign as well? Because F is in opposite direction to r? I read a post online that said if we state -dr we are already assuming the motion of the body is downwards so the limits should be switched and if we state dr then only we...

I am still not clear about the integration because the work done by a variable force is ∫F.dx where dx is a very small DISPLACEMENT. Whereas while calculating the work done by the gravitational force we do ∫F.dr where dr is a radial displacement that is always outwards. So is dr equal to dx? I...

If dr is negative in case of a falling object, the integration gives a negative result since the lower limit will be greater than the upper limit. So, what's wrong?

"dr" is used when potential energy is derived from integration. Here is dr a small displacement in the direction of r? To calculate the work done by gravity using integration when a body if falling is dr positive or negative?

What does the "r" in the formula F = - GMm/r^2 mean? Does it mean distance between the two bodies or the radial displacement ? Is the "r" a vector or a scalar?

Here when you said "when we push it back with the same magnitude, you do F x D work again" I think we should also account the direction of the force because the force is constant both in magnitude and direction throughout the motion. So shouldn't the work done by the force on the crate when...

Yes I have.

In a closed loop when we apply an applied force on an object the object starts at point A and stops at point A. Since the displacement is 0, Work done by the applied force on the object is = F x s x cosθ...

is GPE at a point the work we must do against the gravitational force to bring an object from infinity to the point? Or is it the work done by the gravitational force?

Here you said the system gains energy because Earth also does negative work on the mass.

So that means the gain in potential energy of the earth-mass system should when we lift an object should be 2mgh. I know I am missing something. Could you clarify me?

I meant the work done by Earth not work done on earth.

But what about the negative work done by the Earth on the mass when the mass is lifted? Shouldn't that decrease the energy of the earth-mass system?

What I meant was: Consider a particle on the ground. This particle is raised by a force of magnitude mg to a height h above the ground. At this point, the work done on the particle by the force is mgh, which is equal to the potential energy of the particle. But, during this period, the force of...

This question may sound weird but when we lift an object with a force equal to the weight of the object up to a certain height the Earth is doing negative work on the object as well. So shouldn't the net work be zero?

Why is infinite kept as a lower limit in the derivation of gravitational potential energy? Shouldn't the lower limit be smaller than the upper limit? http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

Why can ratios be written as fractions? if there are 5 men in 20 people the ratio of men to total people is 5:20 which is 1:4. But when we write 1/4 it gives 0.25. And 0.25 means the magnitude of each part when 1 is divided into 4 equal parts. So how does ratio and fraction give a...

Why does "per" in 3 miles per hour mean division? Why are gradients and rates a ratio?

velocity = distance traveled / time taken

Did the formula for velocity come from unitary method?

But doesn't the potential of A and B increase as A or B gets closer because both of the particles a simultaneously changing their positions in the elctric field of each other?

If a positive charge A moves towards another stationary positive charge B then the A's electric potential energy increases. But shouldn't the electric potential energy of B also increase as it is also in a way moving towards the A inside the A's electric field? So shouldn't the total...

V = ωr V= velocity ω = angular velocity r = radius How is the direction of ω perpendicular to the plane of rotation? Is the formula ω = vsinθ χ 1/r which is the cross product of vsinθ and 1/r the reason why ω is perpendicular to the plane of rotation?

Cross product is used to find the perpendicular vector of two vectors. If there is any two vectors in a plane then there is always a perpendicular vector to both of them. So in circular motion if the motion is horizontal then is there a perpendicular force to the object in circular motion?

What is the work of the earthed line with zero potential between two power sources in the op-amp. What is the normal function of Earth wire except for preventing shock?

When we place a voltmeter across a single battery which is connected to another battery in series, does voltmeter show the emf of the single battery or the sum of emf of both the batteries?

When a positive charge leaves the positive part if the battery it has maximum electric potential energy then as it moves through a wire with a zero resistance the charge is closer to the negative side of the battery. So, while traveling in a wire in a circuit does it lose electric potential...

If the potential difference between point A and point B is 10 Volts, then when a unit positive charge passes from A to B, the charge loses 10 J of energy. But when an electron passes from A to B does it gain energy, because in W = Q . V , Q is negative. I am really confused in potential...

So you mean to say the definition in hyperphysics is wrong?

"F in the definition of potential energy is the force exerted by the force field, e.g., gravity, spring force, etc. The potential energy U is equal to the work you must do against that force to move an object from the U=0 reference point to the position r. The force you must exert to move it...

when a magnet is falling through a solenoid, voltage is produced. Once positive is produced and the when the magnet leaves the solenoid negative voltage is produced. Voltage is the energy lost by a unit charge when traveling from one point to the other. So if there is negative voltage...

If we want to push an electron towards another electron we give equal but opposite force which means we push the electron with the same amount of force that it is being pushed back. Shouldn't both the forces cancel out the electron remain stationary?

If I push a book horizontally across a table I do work. But is energy conserved?

So the worker gains energy but the total mechanical energy is not conserved?

" The potential energy U is equal to the work you must do against that force to move an object from the U=0 reference point to the position r. The force you must exert to move it must be equal but oppositely directed." The above definition is from hyperphysics. U = -GMm/R According to...

Why will an object lose energy when external force does negative work on it?

Then what does negative potential mean?

When Earth pulls a mass with gravitational force why does the energy of the earth-mass system decrease? Isn't work just transfer of energy meaning constant overall energy?

Does negative work done mean loss in potential energy?

If a positive charge is pushed towards another positive charge then the potential energy of the system increases. When negative charge attracts positive charge why doesn't the system gain potential energy? In both cases force is being applied?

Gravitational Potential energy is the work done against the gravity to move a mass from one point to the other. So if a mass was falling down to the earth, how is the potential energy defined?

is the upper limit always greater than the lower limit in integration? what should be the limits if we need to calculate total work done in bringing a mass from infinity to distance r from earth.

Can someone derive the formula to calculate the GPE of a mass which is away from the Earth by considering 0 potential at Earth's surface

So can the potential energy of the rocket can be calculated by the formula (Thrust - Gravitational force) * distance In my book the gain in GPE of the rocket is calculated by -GMm/r which basically came from integrating Gravitational force* distance Is there any difference? I am really...

In this case of rocket launch what force is doing work?