Interpreting a one-dimensional potential energy diagram allows you to obtain qualitative, and some quantitative, information about the motion of a particle.
At a turning point, the potential energy equals the mechanical energy and the kinetic energy is zero, indicating that the direction of the velocity reverses there.
The negative of the slope of the potential energy curve, for a particle, equals the one-dimensional component of the conservative force on the particle. At an equilibrium point, the slope is zero and is a stable (unstable) equilibrium for a potential energy minimum (maximum).
Problems
A mysterious constant force of 10 N acts horizontally on everything. The direction of the force is found to be always pointed toward a wall in a big hall. Find the potential energy of a particle due to this force when it is at a distance
x from the wall, assuming the potential energy at the wall to be zero.
10
x with
x -axis pointed away from the wall and origin at the wall
A particle of mass 4.0 kg is constrained to move along the
x -axis under a single force
where
The particle’s speed at
A , where
is 6.0 m/s. What is its speed at
B , where
The force on a particle of mass 2.0 kg varies with position according to
(
x in meters,
F (
x ) in newtons). The particle’s velocity at
is 5.0 m/s. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b)
as the reference point. (c) Find the particle’s velocity at
Do this part of the problem for each reference point.
A 4.0-kg particle moving along the
x -axis is acted upon by the force whose functional form appears below. The velocity of the particle at
is
Find the particle’s speed at
Does the particle turn around at some point and head back toward the origin? (e) Repeat part (d) if
A particle of mass 0.50 kg moves along the
x -axis with a potential energy whose dependence on
x is shown below. (a) What is the force on the particle at
12 m? (b) If the total mechanical energy
E of the particle is −6.0 J, what are the minimum and maximum positions of the particle? (c) What are these positions if
(d) If
, what are the speeds of the particle at the positions listed in part (a)?
(a) Sketch a graph of the potential energy function
where
are constants. (b) What is the force corresponding to this potential energy? (c) Suppose a particle of mass
m moving with this potential energy has a velocity
when its position is
. Show that the particle does not pass through the origin unless
a. where
; b.
; c. The potential energy at
must be less than the kinetic plus potential energy at
or
Solving this for
A matches results in the problem.
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