Written Test Principles of Motion
June 4, 2025
1. In a race of speed ` on flat d = 100 meters, two athletes arrive at the finish line
simultaneously, both with a time of T = 10.0 s. The motion of the two athletes can be
approximated as the sum of uniformly accelerated motion in the first segment plus
uniform rectilinear motion until the finish line. The first athlete accelerates for d1 = 20
m, while the second accelerates for d2 = 15 m. Determine:
(a) the acceleration of the two athletes; (b) their speed at the finish line; (c) who is first
after 50 m; (d) who would have won had the race continued for another 10 m and by
what time difference.
2. A body with a mass of m = 60 Kg slides down a smooth inclined plane of θ = 5°; it
starts with zero speed and covers a distance of d = 4 m along the plane. At the end of the
plane, it moves along a smooth horizontal distance of h = 2 m and hits a spring of r
esting length x0 = 0.5m, fixed to a wall. Calculate
(a) the elastic constant of the spring so that ` the body touches the wall with velocity `
equal to zero; (b) the elastic constant so that ` the body touches the wall with velocity `
equal to zero in the case where there is a friction coefficient µd = 0.14 in the
horizontal section; (c) the friction coefficient in the horizontal section necessary so
that ` the body arrives at the spring with velocity ` equal to zero.
x0
h
3. On a smooth circular guide of radius r = 1.0 m, situated in a horizontal plane, there
are two masses m1 = 0.20kg and m2 = 1.0 kg, free to slide along the guide. They are
held by two stops in the position illustrated in the figure, in such a way as to compress a
spring of negligible mass and elastic constant k = 4.0 · 102 N/m, to which they are simply
resting. At the moment t = 0 the stops are removed, and the two masses are launched i
n opposite directions by the expansion of the spring. The initial distance between the t
wo masses and the time taken by the spring to accelerate them are neglected.
(a) Discuss, providing reasoning for your response, whether the conservation of
momentum and angular momentum hold true between the two instants before and
after the spring is released. (b) At what angle θ will the two masses collide? (c) If the
spring was compressed by a distance of x0 = 5 cm, at what instant t will the
collision occur? (d) Assuming the collision is perfectly elastic, at what angle θ′ will
the masses meet for the second time?
1
θ
4. A student pulls on a rope that passes over a pulley of mass M1 = 1 Kg, with an inner radius of r = 5 c
m and an outer radius of R1 = 10 cm, as shown in the figure. The pulley ` is in contact with a disk of
mass M2 = 2 Kg and radius R2 = 15 cm. The disk and pulley are free to rotate without friction around
an axis passing through their respective centers, while there is friction between their edges with a
coefficient of µs = 0.8. Under the assumption that they roll without slipping against each other, and that
the student pulls with a constant force of magnitude F = 10 N, calculate
(a) their angular accelerations α1 and α2; (b) the minimum value of the force that
compresses the two disks against each other so that ´ and sliding does not occur. (c)
Discuss whether and how the angular accelerations change in the case where the
cord is not pulled by the student but by a body attached to the rope, which falls under
the action of gravity, with a mass such that mg = F .
Solution: The second cardinal law for the two bodies is
I1 α1 = F r − Fa R1
e
I2 = α2 = Fa R2
with the constraint that α1 R1 = α2 R2. We can thus eliminate Fa = I2 α2 /R2 and find
one of the two angular accelerations as
α2 =
Fr
2F R
=
= 22.2 rad/s2
I1 R2 /R1 + I2 R1 /R2
(M1 + M2 )R1 R2
from
which
α1 = α2 R2 /R1 = 33.3 rad/s2 .
We can therefore derive the friction force as
Fa =
1
M2 α2 R2 = 3.33N < N µs
2
thus N > Fa /µs = 4.16N.
2