More Applications of the Energy Principle
Objective: To calculate the gravitational potential energy of an object near the surface of the Earth; to
apply the energy principle to a “particle” moving near the surface of the earth; to calculate the potential
energy of two charged particles; to apply the energy principle to a system of charged particles.
Gravitational potential energy for a particle near the surface of the Earth
Near the surface of the Earth, the gravitational potential energy of a particle-Earth system is approximately
Ugrav ≈ mgy
(1)
where g = 9.8 N/kg and y is the height of the particle above the surface of the Earth. It is positive;
however, it has the same features as we expect for gravitational potential energy–it is large for large separation
and it is small for small separation. Thus, it is directly proportional to the separation of the particles (particle
and Earth).
Electric potential energy
What if two charged particles are interacting? Their gravitational interaction is negligible compared to their
electrical interaction. The force of one charged particle on another has a magnitude that is proportional
to the charges of the two particles and inversely proportional to their separation. This force is called the
Coulomb force, or electric force.
|F~electric | =
1 |q1 q2 |
4π0 r2
where
1
= 9 × 109 N m2 /coulomb2
4π0
(2)
It’s direction depends on the signs of the charges. Like charges repel and unlike charges attract. The
potential energy of a system of two charged particles is
Ue =
1 q1 q2
4π0 r
(3)
Note that if the two particles have like charges, they repel and Ue is positive–just as we expect!. If they
attract (unlike charges), Ue is negative–just as we expect!
Frequently electric potential energies are reported in units of electron-volts (eV) where 1eV= 1.6 × 10− 19
J.
Application
1. You toss a 0.050-kg ball vertically into the air. Its speed when leaving your hand is 10.0 m/s.
(a) If the ball is originally at rest in your hand, what is the work done by your hand if the ball’s
height while in your hand changes by 0.50 m?
(b) What will be the maximum height of the ball relative to the height at which it is released if air
resistance is negligible?
(c) Suppose that because of air resistance the ball only travels to a height of 4.5 m. What is the
change in the energy of the ball-earth system during the interval between when the ball leaves
your hand and when it reaches its maximum height?
2. You throw a ball with an initial velocity of magnitude 8.0 m/s (when it leaves your hand) at an angle
of 20◦ .
(a) What will be its maximum height above where it is released?
(b) How will be the speed of the ball when it is at its maximum height?
(c) If you throw another ball with the same speed, but at an angle of 40◦ , which ball will reach a
greater height?
(d) Which ball has the greater speed when it is at its maximum height?
3. A roller coaster of mass 300.0 kg has a speed of 5.00 m/s at the top of the first hill. It descends 15.0
m to the bottom of the hill and then ascends 10.0 m to the top of the second hill.
(a) If the energy of the roller coaster-earth system is conserved, what is the speed of the roller coaster
at the bottom of the first hill?
(b) What is its speed at the top of the second hill?
(c) Suppose that the roller coaster is only going at 7.5 m/s at the top of the second hill. What is
the change in the energy of the system during the interval between the top of the first hill and
the top of the second hill? Express this in terms of joules and also as a percentage of the total
mechanical energy at the top of the first hill (assuming Ugrav = 0 at the bottom of the first hill).
Where did this energy go?
4. A 2-mm-diameter plastic bead is charged to -1 nC (adapted from Randall Knight’s Five Easy Lessons).
(a) An alpha particle (He nucleus) is fired at the bead from far away with a speed of 1 × 106 m/s, and
it collides head-on. What is its speed at impact?
(b) Suppose an electron is fired at the bead from far away and it ”reflects” at a distance of 0.1 mm
from the surface. What was the electron’s initial speed?
5. A charged sphere, shown in Figure ?? has a net positive charge of 1 nC.
Figure 1:
(a) What is the speed of a proton at point 2 if it is moving to the right with a speed of 4 × 105 m/s
at point 1?
(b) What is the speed of a proton at point 2 if it is moving to the left with a speed of 4 × 105 m/s at
point 1?
(c) What is the speed of an electron at point 2 if it is moving to the right with a speed of 4 × 105 m/s
at point 1?