Kevin Zhou Physics Olympiad Handouts
Preliminary Problems
These problems should be approachable, after some thought, if you understand the core material
in Halliday, Resnick, and Krane. If you can solve at least 75% of these completely and correctly,
you’re ready to start the main problem sets. Take care: many of the problems are more subtle
than they look. Answers are provided for most questions, so you can check your work; solutions are
deliberately not provided, so that you have the chance to work them out for yourself.
1 Mechanics
These problems can be solved using the material in chapters 1 through 17 of Halliday and Resnick.
[1] Problem 1. A projectile is thrown up and passes point A, then point B a height h above A. Let
TA and TB be the time intervals between the two times the projectile passes A and B, respectively.
(a) Show that an experimentalist can measure g by computing g = 8h/(TA2 − TB2 ).
(b) This procedure probably looks a little contrived. Why is it better than doing something
simpler, such as just dropping the ball and using ∆y = gt2 /2?
[2] Problem 2. A projectile is launched with speed v at an angle θ from the horizontal on a flat plane.
(a) Find y(x) and the ratio of the range to the maximum height.
(b) What is the maximum θ for which the projectile always increases its distance from the thrower?
√
Answer. (b) sin−1 (2 2/3)
[1] Problem 3. Consider the two following setups involving pulleys and spring scales. Treat the ropes
and spring scales as massless, and the pulleys as frictionless. Model the pulleys as uniform discs
with masses of 5 kg which are fixed by a rigid support.
(a) Draw a free-body diagram for the second setup, showing all external forces on the spring scale.
(b) What are the readings on the two spring scales?
(c) Draw a free-body diagram for the first setup, showing and naming all external forces on the
pulley. In particular, what is the magnitude of the force between the pulley and the rope?
(d) What is the magnitude of the force that must be provided by the support?
√
Answer. (c) normal force from the rope, weight, and the force from the support, (d) 50 5 N
1
, Kevin Zhou Physics Olympiad Handouts
[1] Problem 4. A sewage worker is using a ladder inside a large, frictionless, horizontal circular
aqueduct. The ladder is of the same length as the diameter of the aqueduct.
(a) First the ladder is placed perfectly vertically and the worker climbs to the midpoint. Draw a
free body diagram indicating all forces on the ladder, and their names. Do the forces balance?
(b) Now suppose the ladder is placed perfectly horizontally and the worker hangs statically from
the midpoint. Draw a free body diagram indicating all forces on the ladder, and their names.
Do the forces balance? If so, show this explicitly. If not, what happens next?
[1] Problem 5. A student proposes to build this set of pulleys and string, called a “fool’s tackle”.
If the load L has mass m, and the pulleys and string are frictionless and have negligible mass, find
the force F needed to keep the system static.
Answer. This system can’t be static. It just instantly falls apart, no matter what F is.
[2] Problem 6. A wooden isosceles right triangle with uniform mass density is placed on a table, and
a force is applied as shown.
The force is gradually increased until the triangle begins to tip over without sliding. The force is
then removed. Next, the surface is inclined with angle θ. For what range of θ can you be certain
the triangle will not slide down the incline?
Answer. θ < tan−1 (1/3)
[1] Problem 7. A painter of mass M stands on a platform of mass m as shown.
2
Preliminary Problems
These problems should be approachable, after some thought, if you understand the core material
in Halliday, Resnick, and Krane. If you can solve at least 75% of these completely and correctly,
you’re ready to start the main problem sets. Take care: many of the problems are more subtle
than they look. Answers are provided for most questions, so you can check your work; solutions are
deliberately not provided, so that you have the chance to work them out for yourself.
1 Mechanics
These problems can be solved using the material in chapters 1 through 17 of Halliday and Resnick.
[1] Problem 1. A projectile is thrown up and passes point A, then point B a height h above A. Let
TA and TB be the time intervals between the two times the projectile passes A and B, respectively.
(a) Show that an experimentalist can measure g by computing g = 8h/(TA2 − TB2 ).
(b) This procedure probably looks a little contrived. Why is it better than doing something
simpler, such as just dropping the ball and using ∆y = gt2 /2?
[2] Problem 2. A projectile is launched with speed v at an angle θ from the horizontal on a flat plane.
(a) Find y(x) and the ratio of the range to the maximum height.
(b) What is the maximum θ for which the projectile always increases its distance from the thrower?
√
Answer. (b) sin−1 (2 2/3)
[1] Problem 3. Consider the two following setups involving pulleys and spring scales. Treat the ropes
and spring scales as massless, and the pulleys as frictionless. Model the pulleys as uniform discs
with masses of 5 kg which are fixed by a rigid support.
(a) Draw a free-body diagram for the second setup, showing all external forces on the spring scale.
(b) What are the readings on the two spring scales?
(c) Draw a free-body diagram for the first setup, showing and naming all external forces on the
pulley. In particular, what is the magnitude of the force between the pulley and the rope?
(d) What is the magnitude of the force that must be provided by the support?
√
Answer. (c) normal force from the rope, weight, and the force from the support, (d) 50 5 N
1
, Kevin Zhou Physics Olympiad Handouts
[1] Problem 4. A sewage worker is using a ladder inside a large, frictionless, horizontal circular
aqueduct. The ladder is of the same length as the diameter of the aqueduct.
(a) First the ladder is placed perfectly vertically and the worker climbs to the midpoint. Draw a
free body diagram indicating all forces on the ladder, and their names. Do the forces balance?
(b) Now suppose the ladder is placed perfectly horizontally and the worker hangs statically from
the midpoint. Draw a free body diagram indicating all forces on the ladder, and their names.
Do the forces balance? If so, show this explicitly. If not, what happens next?
[1] Problem 5. A student proposes to build this set of pulleys and string, called a “fool’s tackle”.
If the load L has mass m, and the pulleys and string are frictionless and have negligible mass, find
the force F needed to keep the system static.
Answer. This system can’t be static. It just instantly falls apart, no matter what F is.
[2] Problem 6. A wooden isosceles right triangle with uniform mass density is placed on a table, and
a force is applied as shown.
The force is gradually increased until the triangle begins to tip over without sliding. The force is
then removed. Next, the surface is inclined with angle θ. For what range of θ can you be certain
the triangle will not slide down the incline?
Answer. θ < tan−1 (1/3)
[1] Problem 7. A painter of mass M stands on a platform of mass m as shown.
2
