Concept #1: Relationship Between Force and Potential Energy (In 1-Dimension)

Concept #2: Partial Derivatives

Concept #3: General Relationship Between Force and Potential Energy

A potential energy function is given by
U = Ax3 + Bx2 - Cx + D
Find the stable equilibrium position for A = 3.45 J/m 3, B = 1.05 J/m 2, C = 1.7 J/m, and D = 4.2 J.

The figure shows a potential energy curve, U(x).
At which point does the force have greatest magnitude?For each labeled point, state whether the force acts to the left or to the right, or is zero.Where is there equilibrium and of what type is it?

A marble moves along the x-axis. The potential-energy function is shown in the figure .At which of the labeled x-coordinates is the force on the marble zero?Which of the labeled x-coordinates is a position of stable equilibrium?Which of the labeled x-coordinates is a position of unstable equilibrium?

A particle has potential energy
U( x ) = x + sin( ( 2 )x )
as a particle moves over the range
0 x m.Where are the equilibrium positions in this range?Is x = 3 m a point of stable or unstable equilibrium?Is x=2 3 m a point of stable or unstable equilibrium?

A particle moves along the x-axis while acted on by a single conservative force parallel to the x-axis. The force corresponds to the potential-energy function graphed in the figure . The particle is released from rest at point exttip{A}{A}.What is the direction of the force on the particle when it is at point exttip{A}{A}?At point exttip{B}{B}?At what value of exttip{x}{x} is the kinetic energy of the particle a maximum?What is the force on the particle when it is at point C?What is the largest value of exttip{x}{x} reached by the particle during its motion?What value or values of exttip{x}{x} correspond to points of stable equilibrium?Of unstable equilibrium?

A particle with mass m is acted on by a conservative force and moves along a path given by x = x0 0 t and y = y0 0 t, where x0, y0, and 0 are constants.Find the components of the force that acts on the particle.Find the potential energy of the particle as a function of exttip{x}{x} and exttip{y}{y}. Take U = 0 when x = 0 and y = 0.Find the total energy of the particle when x = x0, y = 0.Find the total energy of the particle when x = 0, y = y0.

A particular spring obeys the force law = ( - k x + a x3 + b x4 ) .Is this force conservative?Determine the form of the potential energy function U(x). Assume U(0) = 0.

A particle is constrained to move in one dimension along the exttip{x}{x} axis and is acted upon by a force given by large{vec{f F}(x)=-frac{k}{x^3}hat{f i}} where exttip{k}{k} is a constant with units appropriate to the SI system.Find the potential energy function U(x), if exttip{U}{U} is arbitrarily defined to be zero at x = 2.0 m, so that U(2.0) = 0.

A particle constrained to move in one dimension is subject to a force F(x) that varies with position x as (x) = A(kx) where exttip{A}{A} and exttip{k}{k} are constants.What is the potential energy function U(x), if we take U = 0 at the point x = 0?

A clever engineer designs a "sprong" that obeys the force law Fx=-q(x-xe)3, where xe is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, well let xe=0. Then
Fx=-qx3.Find an expression for the potential energy of a stretched or compressed sprong.A sprong-loaded toy gun shoots a 21 g plastic ball. What is the launch speed if the sprong constant is 4.3×104 N/m3 and the sprong is compressed 13 cm ? Assume the barrel is frictionless.What are the units of q?

A 100 g
particle experiences the one-dimensional, conservative force Fx shown in the figure.Let the zero of the potential energy be at
x = 0 m.What is the
potential energy at
x = 1.0, 2.0, 3.0, and 4.0 m? Hint: Think about the definition of potential energy and the geometric interpretation of the work done by a varying force.Suppose the particle is shot toward the right from
x = 1.0 m
with a speed of
24
m/s. Where is the particles turning point?

A certain spring is found not to obey Hookes law; it exerts a restoring force Fx(x)= - x- x2 if it is stretched or compressed, where = 60.0 N/m and = 18.0 N/m2. The mass of the spring is negligible.Calculate the potential energy function U(x) for this spring. Let U=0 when x=0.An object with mass 0.900 kg on a frictionless, horizontal surface is attached to this spring, pulled a distance 1.00 m to the right (the + x - direction) to stretch the spring, and released. What is the speed of the object when it is 0.50 m to the right of the x=0 equilibrium position?

A proton with mass m moves in one dimension. The potential-energy function is U( x ) = /x2 - /x, where 0 = / 0 = / and eta are positive constants. The proton is released from rest at x_0 = alpha / eta.Show that U( x ) can be written as
large{Uleft( x
ight) = frac{ alpha }{{x_0 ^2 }}left[ {left( {frac{{x_0 }}{x}}
ight)^2 - frac{{x_0 }}{x}}
ight]}Calculate v( x ), the speed of the proton as a function of position.For what value of exttip{x}{x} is the speed of the proton a maximum?What is the force on the proton at the point in part D?Let the proton be released instead at x1 = 3 / 1 = 3 / ( x ). Calculate vleft( x
ight).For the release point x = x0, what are the maximum and minimum values of exttip{x}{x} reached during the motion?Calculate U( x0 ).What is the value of that maximum speed?For the release point x = x1, what are the maximum and minimum values of exttip{x}{x} reached during the motion?

A small object with mass
m = 0.0900 kg
moves
along the +x-axis. The only force on the object is a conservative
force that has the potential-energy function U(x) = -x2 + x3,
where
=
6.00 J/m2
and
J/m3 = 0.300 x.
The object is released
from rest at small x.When the object is at
x = 4.00 m, what is its speed?When the object is at
x = 4.00 m, what is the magnitude of its acceleration?When the object is at
x = 4.00 m, what is the direction of its acceleration?
What is
the maximum value of x reached by the object during its motion?

A conservative force vec{F} is in the
+x-direction and has magnitude
F(x) = /(x+x0)2, where
exttip{alpha }{alpha} = 0.600 Nm2
and
x0 = 0.200 m.
What is the potential-energy function U(x) for this force? Let
U(x) 0
as x.An object with mass
m = 0.500 kg
is released from rest at
x = 0 and moves in the
+x-direction. If vec{F} is the only force acting on the object,what is the objects speed when it reaches
x = 0.400 m?

shows the potential energy of a 500 g
particle as it moves along the
x-axis. Suppose the particles mechanical energy is 12 J.Where are the particles turning points?What is the particles speed when it is at
x = 4.0 m?What is the particles maximum speed?At what position or positions does this occur?Suppose the particle’s energy is lowered to 4.0 J. Can the
particle ever be at
x = 2.0 m? Suppose the particle’s energy is lowered to 4.0 J. Can the
particle ever be at
x = 4.0 m?

A system in which only one particle can move has the potential energy shown in
. Suppose
U1 = 16
J.
You may want to review (Pages 248 - 249).What is the
x-component of the force on the particle at
x = 5 cm?What is the
x-component of the force on the particle at
x = 15 cm?What is the
x-component of the force on the particle at
x = 25 cm?What is the
x-component of the force on the particle at
x = 35 cm?

In a particle has a mass of
220
g.What is the maximum speed the particle could have at
x = 2.0 m
and never reach
x = 6.0 m?

shows the potential energy of a system in which a particle moves along the
x-axis.Draw a graph of the force Fx as a function of position x for the x(0,0.5).Draw a graph of the force Fx as a function of position x for the x(0.5,1).

is the potential-energy diagram for a 20 g
particle that is released from rest at
x = 1.0 m.What is the particles maximum speed?Where are the turning points of the motion?Will the particle move to the right or to the left?At what position does it have this speed?

A particle in has a mass of 100 g.What minimum speed does the particle need at point A to reach point B?What minimum speed does the particle need at point B to reach point A?

What is the maximum speed of a 2.0 g
particle that oscillates between
x =2.0 mm
and
x =8.0 mm
in
?

A system in which only one particle can move has the potential energy shown in
. Suppose
U1 = 80
J.
What is the
y-component of the force on the particle at
y = 0.5 m
?What is the y-component of the force on the particle at
y = 4 m?

Two point masses, m1 and m2, lie on the x-axis, with m1 held in place at the origin and m2 at position exttip{x}{x} and free to move. The gravitational potential energy of these masses is found to be U( x) = - Gm1 m2 /x, where exttip{G}{G} is a constant (called the gravitational constant).Find the x-component of the force acting on m2 due to m1.Is this force attractive or repulsive?How do you know?

If U = 3x2 + 2xy + 4y2 z, what is the force, vec{f F}?

The potential energy of a pair of hydrogen atoms separated by a large distance exttip{x}{x} is given by U( x ) = - C6 /x6, where C6 is a positive constant.What is the force that one atom exerts on the other?Is this force attractive or repulsive?

An object moving in the xy-plane is acted on by a conservative force described by the potential-energy function U(x, y)= (1 / x2+1 / y2), where alpha is a positive constant.Derive an expression for the force vec{F} expressed in terms of the unit vectors i and j.

A small block with mass 0.0400 kg is moving in the xy-plane. The net force on the block is described by the potential- energy function U(x,y)= (5.80 J/m2 )x2-(3.70 J/m3 )y3.
What is the magnitude of the acceleration of the block when it is at the point x=
0.31 m , y=
0.61 m ?What is the direction of the acceleration of the block when it is at the point x=
0.31 m , y=
0.61 m ?

The potential energy for a particle that can move along the x-axis is U=Ax2 + B(x /L), where A, B, and L are constants.What is the force on the particle at
x=0?What is the force on the particle at
x=L/2?What is the force on the particle at x=L?

A particle moving along the
y-axis has the potential energy
U=6y3, where y is in m.What is the y-component of the force on the particle at
y = 0 m?What is the y-component of the force on the particle at
y = 1 m?What is the y-component of the force on the particle at
y = 2 m?

A particle moving along the
x-axis is in a system with potential energy
U=16/x, where x is in m.What is the
x-component of the force on the particle at
x = 2 m?What is the
x-component of the force on the particle at
x = 5 m?What is the
x-component of the force on the particle at
x = 8 m?

The following graph represents a hypothetical potential energy curve for a particle of mass m.If the particle is released from rest at position r 0, its speed ||→v|| at position 2r0 is most nearly

The plot shown represents the potential energy of a particle that can only move along the x-axis, as a function of the coordinate x. Which of the following statements is correct:[a] In B and D the force is zero;[b] The force in B is positive;[c] A and C are stable equilibrium positions;[d] The force in D is positive;[e] A and C are unstable equilibrium positions.

The potential energy between two neutral atoms is called the Lennard-Jones potential, and is given by the equation
U(x) = A/r12 - B/r6
where A and B are positive constants.
a) What is the force due to this potential energy?
b) For what range of r is the force attractive, and for what range of r is the force repulsive? Note that if the force is positive, it is repulsive, and if the force is negative, it is attractive.

A force parallel to the x-axis acts on a particle moving along the x-axis. This force is associated with potential energy U(x) given by U(x) = αx4, where αexttip{alpha }{alpha} = 0.710 J/m4.(a) What is the magnitude of the force when the particle is at x = -0.800 m?(b) What is the direction of the force when the particle is at x = -0.800 m?

A single conservative force acts on a 5.00-kg particle within a system due to its interaction with the rest of the system. The equation Fx = 2x + 4 describes the force, where Fx is in newtons and x is in meters. As the particle moves along the x axis from x = 1.00 m to x = 5.00 m, calculate(a) the work done by this force on the particle(b) the change in the potential energy of the system(c) the kinetic energy the particle has at x = 5.00 m if its speed is 3.00 m/s at x = 1.00 m

A particle that can move along the x-axis experiences an interaction force
Fx=(3x2-5x) N where x is in m. Find an expression for the system's potential energy.

The following image is the potential-energy diagram for a 500 g particle that is released from rest at A.(a) What is the particles speed at B?(b) What is the particles speed at C?(c) What is the particles speed at D?

Starting from rest, a 64.0-kg person bungee jumps from a tethered hot-air balloon 65.0 m above the ground. The bungee cord has negligible mass and unstretched length 25.8 m. One end is tied to the basket of the balloon and the other end to a harness around the person’s body. The cord is modeled as a spring that obeys Hooke’s law with a spring constant of 81.0 N/m, and the person’s body is modeled as a particle. The hot-air balloon does not move. (a) Express the gravitational potential energy of the person–Earth system as a function of the person’s variable height y above the ground. (b) Express the elastic potential energy of the cord as a function of y. (c) Express the total potential energy of the person–cord–Earth system as a function of y. (d) Plot a graph of the gravitational, elastic, and total potential energies as functions of y. (e) Assume air resistance is negligible. Determine the minimum height of the person above the ground during his plunge. (f) Does the potential energy graph show any equilibrium position or positions? If so, at what elevations? Are they stable or unstable? (g) Determine the jumper’s maximum speed.

Name the type of equilibrium for each position of the balls in the figure.

For the potential energy curve shown in Figure P7.52, (a) determine whether the force Fx is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for Fx versus x from x = 0 to x = 9.5 m.

A particle that can move along the x-axis is part of a system with potential energy where A and B are positive constants.(a) Where are the particles equilibrium positions in this range?(b) For each, s it large{{
ormalsize x=}frac{2A}{B}}a point of stable or unstable equilibrium?

A particle of mass m moves under the influence of a potential energy large{U(x) = frac{a}{x} + bx}, where a and b are positive constants and the particle is restricted to the region x > 0. Find a point of equilibrium for the particle and demonstrate that it is stable.

A single conservative force acting on a particle within a system varies as , where A and B are constants, is in newtons, and x is in meters. (a) Calculate the potential energy function U(x) associated with this force for the system, taking U = 0 at x = 0. Find (b) the change in potential energy and (c) the change in kinetic energy of the system as the particle moves from x = 2.00 m to x = 3.00 m.

(a) Take U = 5 for a system with a particle at position x = 0 and calculate the potential energy of the system as a function of the particle position x. The force on the particle is given by . (b) Explain whether the force is conservative or nonconservative and how you can tell.

The potential energy of a system of two particles separated by a distance r is given by U(r) = A/r, where A is a constant. Find the radial force that each particle exerts on the other.

A potential energy function for a system in which a two-dimensional force acts is of the form U = 3x3y - 7x. Find the force that acts at the point (x, y).