Problem Set 4: Problems in Translational Mechanics
In this problem set, we go through more problems in translational mechnanics using all of the
ideas we have learnt so far — forces, energy, and momentum. The idea of using non-inertial
frames, where we need to consider fictitious forces, will also make an apperarance.
A CONSCIOUS NOTE
Make sure you understand the physical intuition for each question as the very first
step. Visualise the situation and see how all the particles are moving around, what is
under the influence of what force / potential, who is interacting with who, et cetera.
Then, think about what approach is most appropriate for doing the question — Force
analysis? Energy conservation? Momentum conservation? Or a combination of two of
them? Or the combination of all three?
Only then, when you’re sure of your physical intuition and idea, think about how to
use mathematics to write down the equations and solve them. The mathematics must be
motivated by the physics, not the other way around! Make sure you consciously remind
yourself of this every once in a while until it becomes a habit.
1 An ideal spring of negligible mass was used to hang a load vertically to the ceiling.
Then, the spring as well as the load it was carrying are cut into two identical pieces, and
arranged in the manner shown in the diagram below.
In which setup, before or after, is the lowest point of the bottom mass lower? (The
diagram clearly depicts the “after” case to be lower, but the diagram may not be drawn
to scale.) Give a qualitative argument, as well as a quantitative one.
2 (Isaac Physics, adapted) Two identical springs of spring constant k are used to hold up
a mass m in a triangular fashion, shown below. The distance btween the two points of
support at the ceiling is d, the relaxed length of each spring is negligible, and the springs
hang at angle β to the vertical as shown in the diagram. Gravitational acceleration is g.
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(a) Determine β in terms of the other given parameters at the equilibrium position.
(b) Find the angular frequency of small oscillations of the ball when given a small
vertical displacement. Do this in two ways: using forces, and using potential energy.
3 Two identical masses m are connected horizontally by a spring of relaxed length L and
spring constant k, and allowed to move horizontally on a frictionless surface.
The system is initially at rest, but the mass on the left is given a sudden impulse p.
Describe the subsequent motion of the system, including all qualitative and quantitative
information that may be of interest. Be as thorough as you can be in your account —
imagining as though you were a question setter, set as many difficult questions as you
can about this system for yourself and ensure you can answer them all.
How would things change if the two masses were different?
4 Two masses M and m exist in the situation as shown below. An external force F is applied
as shown to the force M horizontally towards the right. The coefficient of friction between
the two masses is µ for both static and kinetic friction.
(a) What is the minimum F necessary to keep m from sliding downwards?
(b) Supposing that F is below the threshold you found previously, and that block m
has uniform density, height h, and starts with its bottom at distance d above the
ground, figure out everything you can about the motion of block m.
(For example: find its speed at height y, speed when it hits the ground, etc.)
5 (Morin) A small mass rests on top of a fixed frictionless sphere. The mass is given a tiny
kick and slides downwards. At what point does it lose contact with the sphere?
6 (Morin) A massless spring of length 2` connects two hockey pucks that lie on frictionless
ice. A constant horizontal force F is applied to the midpont of the string, perpendicular
to it.
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After some time, the pucks are pulled towards the x-axis, then collide and stick together.
The goal of this question is to figure out how much kinetic energy is lost in the collision.
(a) After some time, the two strings form angle θ with the x-axis as shown below.
What is the incremental amount of work done in the y-direction δW y on each of
the pucks? Write the answer either in terms of dy or dθ.
(b) Hence, by integration, determine the value of 12 mvy2 for each of the pucks just before
the collision, and determine the amount of energy lost in the collision.
(c) (Difficult) The answer you obtained should be very slick and nice. Can you find a
simple way to get that result without the need for any integrals?
7 (Morin) For some odd reason, you decide to throw baseballs at a car of mass M that is
free to move frictionlessly on the ground. You throw the balls at the back of the car at
speed u, and they leave your hand at a mass rate of σ kg/s (assume the rate is continuous,
for simplicity). If the car starts at rest, find its speed and position as functions of time,
assuming the balls bounce elastically directly backward off the back window.
Do the problem again, assuming instead that the back window is open so that the balls
collect inside the car.
8 There exist the following rules for calculating the effective spring constant for systems of
more than two spring arranged in series and parallel:
keff = k1 + k2 (parallel),
1 1 1
= + (series).
keff k1 k2
Let us go through the proof of these formulas. Remember that what it means for a
system to have a certain “effective spring constant” is just that it behaves similarly to a
system with only one spring at that spring constant — that is to say, when compressed
by a certain displacement δ, the system responds with a force −keff δ. We may take this
property as the definition of effective spring constant.
(a) Consider two springs arranged in parallel, as below.
Explain why the effective spring constant is just keff = k1 + k2 . Make sure your
explanation is clear and convincing, linking back to the definition of keff above.
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(b) Consider two springs arranged in series now:
By performing force analysis on the system, explain why the effective spring constant
is given by the formula stated above. (This is significantly harder than the parallel
case, and requires you to have some physical insight.)
(c) Extending from your results of the previous part, suppose that we have a spring of
spring constant k and relaxed length L. If I were to cut out a segment of this spring
of length L0 < L, what would be the spring constant of this segment?
(Consdier first chopping the spring into a few identical pieces, so L0 = L/n for some
positive integer n, before extending more generally.)
9 Even though we usually assume the “lab frame” is inertial, labs on the surface of the Earth
are never actually inertial. Give two different reasons why, and figure out a quantitative
way to estimate how big each of these two effects are. Using this, explain why it is
nevertheless an acceptable approximation to say that our “lab frames” are inertial.
10 Three blocks collide as shown in the diagram below. Blocks A and B are connected by
a spring with spring constant k, and are initially stationary. Block C comes in with an
initial speed v0 and hits block A. Neglect friction and air resistance.
Assuming that the collision is perfectly elastic, determine the maximum values of the
following parameters in the subsequent motion of the system:
(a) The contraction of the spring;
(b) The speed of block B.
11 A secondary mass m is orbiting around a much greater primary mass M m in a
circular path at distance d. In so doing, the two ends of the secondary mass closer to
and further away from the primary mass experience slightly different gravitational forces.
This leads to a “tidal force” which appears, from the perspective of the CM frame of the
secondary mass, to pull the secondary mass apart.
If this tidal force exceeds the self-gravity of the secondary mass, then the secondary mass
will disintegrate. The threshold distance when this occurs is known as the Roche limit:
r
ρM
d= 3 2 R
ρm
where R is the radius of the primary mass and ρM , ρm are the densities of the primary
and secondary masses respectively. Prove the Roche limit, making whatever assumptions
are necessary, but make sure you state the assumptions clearly and explicitly.
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12 (2012 USPhT Semifinals) Two identical masses m separated by a distance l are given
initial velocities v0 as shown in the diagram below. The masses interact only through
universal gravitation.
(a) Under what conditions will the masses eventually collide?
(b) Under what conditions will the masses follow circular orbits of diameter l?
(c) Under what conditions will the masses follow closed orbits?
13 Consider a planet m in circular orbit at radius R around a star of mass M , where m M .
There exist five Lagrange points in this orbital configuration, where a much smaller mass
such as an asteroid can remain and exist in a co-orbital configuration with the planet,
having the same orbital period such that the asteroid appears stationary from the planet’s
perspective. The first point L1 lies on a straight line directly between m and M .
L4
L3 Star Planet
L1 L2
L5
The four parts of this question are independent of each other.
(a) Find the distance x of the first Lagrange point L1 from the planet m.
(You will need to make an approximation — be careful in justifying it.)
(b) Explain why the point diametrically opposite the planet in the circular orbit, labelled
L3 in the diagram, is indeed a Lagrange point.
(c) Explain why the two points L4 and L5 , which exist at the third vertices of two
equilateral triangles who respectively have the star and the planet as the other two
vertices, are also Lagrange points.
(d) How would you determine whether each of the Lagrange points is stable?
14 A rod of length 2`, fixed in position at its center, rotates at some constant angular speed
ω. A small bead is free to move along the length of the rod without friction, and begins
at some small displacement `0 from the center of the rod.
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Find the time taken for the bead to reach the end of the rod. Assuming that the end
of the rod has a hole that allows the bead to fly out, find the speed at which it does so
relative to an inertial observer.
15 (Morin) A billiard ball collides elastically with an identical stationary one. Use the fact
that 12 mv 2 can be written as 12 m(v · v) to show that the angle between the resulting
trajectories is 90◦ .
(Hint: take the dot product of the conservation of momentum equation with itself.)
16 (Morin) A pendulum of length L is held with its string horizontal, and then released.
The string runs into a peg a distance d below the pivot, as shown in the diagram. What
is the smallest value of d for which the string remains taut at all times?
17 (Morin) In this question, we consider a simple model for air resistance.
A sheet of mass M moves with speed V through a region of space that contains particles
of mass m and speed v. There are n of these particles per unit volume. The sheet moves
in the direction of its normal. Assume m M , and that the particles do not interact
with each other.
For simplicity, we assume that all the particles are moving with velocity ±v/2 in the
direction of the sheet’s motion. (They may also have some velocity in a direction parallel
to the sheet’s surface, but this doesn’t turn out to be relevant for the question.)
By considering the collisions of the small particles with the sheet, find the drag force per
unit area on the sheet. Consider limiting cases v/V & 0 and v/V % ∞.
18 (Morin) A chain with length L and linear mass density σ lies striaght on a frictionless
horizontal surface. You grab one end and pull it back along itself, in a parallel manner.
Assume that you pull it at constant speed v.
(a) What force must you apply?
(b) What is the total work that you do by the time the chain is straightened out?
(c) How much energy is lost to heat, if any?
