University of South Australia
UniSA: STEM
MATH 3040 Topics in Mathematics 1: Theoretical Mechanics
Assignment #1, SP2 2022
Static Equilibrium and Particle Dynamics in 1D and 2D
This is an individual assessment piece. It is expected that students will submit results of their own working and
that no solution has been devised by a third party. Penalties apply.
Attempt all questions. Marks are as assigned to each question.
Date Due: April 8, 2022, 5pm
Make sure to relate on a coversheet any information about help given/received for any of the questions you
attempted, who received/gave help, and to reference any sources other than the textbook for the course.
Assignment Questions
1. (10 marks)
Three cords are knotted together at a (free-standing) point O. Two of the cords pass over small frictionless
pulleys at A and B on either side of O. The third cord hangs freely under O. The pulleys are separated
horizontally by a distance of 2m, with pulley B positioned 1m lower than the level of pulley A. Hanging from
each cord is a body of weight: w1, w3, and w2 respectively, at A, O and B(!!!). With the system in equilibrium,
the cord section OA makes an angle θ1 with the horizontal, while the cord section OB makes an angle of θ2
with the horizontal.
(a) Draw a diagram representing the set up of weights and pulleys.
(b) Determine the weights w1 and w2 if w3 = 20N, θ1 = 53◦ and θ2 = 37◦.
(c) Determine the weights w1 and w3 if w2 = 20N, θ1 = 30◦ and θ2 = 0◦.
(d) Determine the angles θ1 and θ2 if w1 = 20N, w2 = 16N and w3 = 6N.
(e) Determine angle θ2, weight w1 and weight w2 if w1 = 20N and w3 = 20N, and θ1 = 30◦.
(f) What would happen if w3 = 20N, w2 = 10N, and w1 = 10N?
2. (10 marks)
Using the variation of parameters method (outlined in Lecture 3b) determine the Green’s function for the motion
of a critically damped oscillator. Then using this result, or otherwise, solve for the motion of the oscillator,
initially at rest at position x = x0 6= 0 at t = 0, subjected to the forcing function F (t) = F0(1−e−at). Assume
that k = ma2 and b = 2ma.
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3. (10 marks)
A projectile is fired from the earth’s surface from a gun whose angle of elevation is α and whose muzzle speed
is V . Take the origin to be the muzzle of the gun, the z-axis as the vertical and the x-axis as the horizontal so
that the motion is confined to the xz-plane.
(a) Show that (in the absence of air resistance) the projectiles’s path is given by
z = x tanα− gx
2
2V 2 cos2 α
.
(b) Show that the range of the gun along an inclined plane is given by
L =
2V 2 cosα sin(α− θ)
g cos2 θ
,
where θ is the angle of inclination of the plane. Determine the maximum range achievable (w.r.t. α) on
this inclined plane.
(c) Determine the highest point on a vertical cliff that is within range, say a distance S from the gun.
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