From this equation, you can see why
[link] is the condition for the work to be an exact differential, in terms of the derivatives of the components of the force. In general, a partial derivative notation is used. If a function has many variables in it, the derivative is taken only of the variable the partial derivative specifies. The other variables are held constant. In three dimensions, you add another term for the
z -component, and the result is that the force is the negative of the gradient of the potential energy. However, we won’t be looking at three-dimensional examples just yet.
Force due to a quartic potential energy
The potential energy for a particle undergoing one-dimensional motion along the
x -axis is
where
Its total energy at
and it is not subject to any non-conservative forces. Find (a) the positions where its kinetic energy is zero and (b) the forces at those positions.
Strategy
(a) We can find the positions where
so the potential energy equals the total energy of the given system. (b) Using
[link] , we can find the force evaluated at the positions found from the previous part, since the mechanical energy is conserved.
Solution
The total energy of the system of 2 J equals the quartic elastic energy as given in the problem,
At both positions, the magnitude of the forces is 8 N and the directions are toward the origin, since this is the potential energy for a restoring force.
Significance
Finding the force from the potential energy is mathematically easier than finding the potential energy from the force, because differentiating a function is generally easier than integrating one.
A conservative force is one for which the work done is independent of path. Equivalently, a force is conservative if the work done over any closed path is zero.
A non-conservative force is one for which the work done depends on the path.
For a conservative force, the infinitesimal work is an exact differential. This implies conditions on the derivatives of the force’s components.
The component of a conservative force, in a particular direction, equals the negative of the derivative of the potential energy for that force, with respect to a displacement in that direction.
Conceptual questions
What is the physical meaning of a non-conservative force?
A force that takes energy away from the system that can’t be recovered if we were to reverse the action.
A bottle rocket is shot straight up in the air with a speed
. If the air resistance is ignored, the bottle would go up to a height of approximately
. However, the rocket goes up to only
before returning to the ground. What happened? Explain, giving only a qualitative response.
An external force acts on a particle during a trip from one point to another and back to that same point. This particle is only effected by conservative forces. Does this particle’s kinetic energy and potential energy change as a result of this trip?
The change in kinetic energy is the net work. Since conservative forces are path independent, when you are back to the same point the kinetic and potential energies are exactly the same as the beginning. During the trip the total energy is conserved, but both the potential and kinetic energy change.
A force
acts on a particle as it moves along the positive
x -axis. (a) How much work does the force do on the particle as it moves from
to
(b) Picking a convenient reference point of the potential energy to be zero at
find the potential energy for this force.
A force
acts on a particle. (a) How much work does the force do on the particle as it moves from
to
(b) Picking a convenient reference point of the potential energy to be zero at
find the potential energy for this force.
The potential energy function for either one of the two atoms in a diatomic molecule is often approximated by
where
x is the distance between the atoms. (a) At what distance of seperation does the potential energy have a local minimum (not at
(b) What is the force on an atom at this separation? (c) How does the force vary with the separation distance?
A crate on rollers is being pushed without frictional loss of energy across the floor of a freight car (see the following figure). The car is moving to the right with a constant speed
If the crate starts at rest relative to the freight car, then from the work-energy theorem,
where
d , the distance the crate moves, and
v , the speed of the crate, are both measured relative to the freight car. (a) To an observer at rest beside the tracks, what distance
is the crate pushed when it moves the distance
d in the car? (b) What are the crate’s initial and final speeds
and
as measured by the observer beside the tracks? (c) Show that
and, consequently, that work is equal to the change in kinetic energy in both reference systems.
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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