Download Conservation of Energy Problems

SPH4U: Conservation of Energy Problems
Recap: Work-Energy Theorem
For each of the situations described below, circle 0 work, (+) work, or (-) work being done on the object, and
briefly explain your answer.
Situation #1:
Situation #2
Situation #3
You apply a force of 500 N to push
around a shopping cart at a constant
speed of 1.5 m/s for an hour.
A heavy box of mass (20 kg) slides to a
stop (due to friction) from 3.5 m/s over a
displacement of 12 m.
A waitress carries a 2 kg tray of plates
and glasses at the same height for 35 m
and constant speed.
0 work
+ work
- work
0 work
+ work
- work
0 work
+ work
- work
Conservation of Energy and Energy Bar Charts
An energy bar chart can be a useful tool to visualize the approximate amount of energy stored in each form (Eg, Ek,
elastic energy, etc.) at 2 different points in time (the exact height of the bars is not important). The middle bar in the
chart represents the energy flow into or out of the system by work Wext being done on the system by an external
force (e.g. friction, an applied force, etc.). The bar chart can help us write a conservation of energy equation
relating the total energy of the system at one point in time to the energy at a 2nd point in time.
Case Study: Roller
Coaster Physics
Cate, Beth and Kayleigh
are on the amazing new
Physics-Loop-Of-Death
ride at Canada’s
Wonderland.
Cate notes that the ride is
frictionless and operates
by gravity alone (i.e. Wext
= 0).
Beth measures their
velocity at the bottom of
the loop to be 20 m/s.
Eg
+
Ek
+
Wext = Eg´
+
Ek´
+
0
Kayleigh notices, while
eating cotton candy, that
their car is motionless at the
top.
-
(a) Shade in the bar graphs for each form of energy, with the energies at the bottom of the loop on the left of the
middle bar, and the energies at the top of the loop on the right of the middle bar (or vice versa if you prefer).
(b) Referring to your bar chart, write an equation for the conservation of mechanical energy (Eg + Ek) relating the
energy at the bottom of the loop to the top of the loop. Using this equation, calculate the height of the ride.
(Show that you do not need to know the mass of the car and occupants to do this calculation!)
(c) A little while later, the intrepid trio finds themselves
15 m above the bottom of the loop. Shade in the bar
chart to the right relating the energy at this point of
the ride to the energy at either the bottom or top of
the loop. Then, write an equation for the
conservation of energy, and calculate their speed at
this point in the ride. (Use the height that you found
in part (b).)
Eg
+
Ek
+
Wext = Eg´
+
Ek´
Eg
+
Ek
+
Wext = Eg´
+
Ek´
+
0
-
Case Study: Friction
(Text p.201 - #6) A skier
of mass 55.0 kg slides
down a slope 11.7 m long,
inclined at an angle φ to
the horizontal. The
magnitude of kinetic
friction is 41.5 N. the
skier’s initial speed is 65.7
cm/s and the speed at the
bottom of the slope is 7.19
m/s. Air resistance is
negligible. Determine the
angle φ.
(a) Start by sketching a
diagram, labeling all
the quantities. Express
the height of the slope
using a trig function.
(b) Shade in the bar chart
to the right. Notice
you now have Wext
due to friction – is it
positive or negative?
(c) Write an equation
based on your energy
chart. Solve for angle
φ.
+
0
-
Conservation of Momentum + Conservation of Energy
The drama crew is measuring the speed of bullets
for a production, using a ballistics pendulum (as
shown to the right). Emma, with keen eye and
steady nerves, fires the bullet horizontally into a
block of wood suspended by 2 light strings. The
bullet remains embedded in the wood (i.e. an
inelastic collision) and the wood and bullet
together swing upwards.
The bullet is of mass m with an initial velocity v,
the block is of mass M, and the block and bullet
together swing up to a maximum vertical height h.
This problem has 2 separate events: (1) the bullet colliding with the wood block, and (2) the swing of the pendulum
to its maximum height. The first event is a conservation of momentum problem, and the second event is a
conservation of energy problem.
(a) If the horizontal momentum of the bullet-block system is conserved during the collision, write an equation for
the total momentum of the bullet-block system before and after the collision. Then, derive an algebraic
expression for the speed of the bullet-block immediately after the collision.
(b) As there is no work done by external forces (e.g. friction), use the law of conservation of energy and write an
equation for the bullet-block at the bottom of its swing and at the top of its swing. Then, use this equation and
the expression from part (a) to derive an expression for the initial speed of the bullet v in terms of m, M, g and
h.
(c) If a bullet of mass 8.7 g hits a block of mass 5.212 kg, and the bullet and block swing up to a height of 6.2 cm,
what is the initial speed of the bullet? (6.6×102 m/s)