MECH1
Problem Sheet 1 and Solutions
Due Friday 13 February
Kinematics
Do all problems
1. It was shown in class that if e1,2,3 and e1,2,3 are two orthonormal bases, then for a vector a which has
coordinates (a1 , a2 , a3 ) in the old basis, i.e., a = a1 e1
Mechanics-Calculus Link Problem Sheet and Solutions
Vectors review
1. Find the angle between the vectors a = 3i + j + k, b = 2i 3j k.
Answer:
ab
10
=
.8,
11 14
aa bb
a
The unit vector in the direction of a is a = |a| = a .
11
cos =
The unit vector in th
SAMPLE PROBLEMS FOR MECHANICS 1
The exam will contain 5 short problems, all to be done, and 3 long problems, 2 of which to be
done. Some examples of the short(ish) problems below may, perhaps, be at places a bit longer
than the actual short problems. Some
MECH1
Problem Sheet 1 Solutions
Due Wednesday Feb 6
Kinematics
Do all problems
1. (a) A particle P1 is moving along the positive x axis with the velocity v1 = 150, 000km/s. A particle P2 is
moving in the opposite direction, so that the distance between P1
MECH1
Problem Sheet 2
Due Wednesday Feb 13
Dynamics
Do all problems
1. A particle in the plane is acted upon by a force
F (x, y) = y 2 i + j.
Determine the trajectory of the particle that starts from rest at the origin.
Answer: By the second law of Newton
MECH1
Problem Sheet 8
All problems are due Friday May 9
Constants of motion, angular momentum
1. The net force on a particle moving in three dimensions is F = ai + bj with a, b = 0. Find the direction(s)
in which (i) the momentum, (ii) angular momentum ar
Mechanics-Calculus Link Problem Sheet
Vectors
1. Find the angle between the vectors a = 3i + j + k, b = 2i 3j k. Normalize these vectors. Find the
length of the projection of a on the direction of b and of b on the direction of a.
2. Show that a is perpen
MECH1
Problem Sheet 5 Solutions
Work and Energy, continued
Do all problems
1. (This is the last problem from the previous homework) The one-dimensional potential energy function for
1
1
a particle is given as U (x) = x3 x . Sketch U , identify and describ
MECH1
Problem Sheet 6 Solutions
Constants of motion, angular momentum
1. The net force on a particle moving in three dimensions is F = ai + bj with a, b = 0. Find the direction(s)
in which (i) the momentum, (ii) angular momentum are conserved.
Answer: Sin
MECH1
Problem Sheet 4
Due Friday March 13
Force impulse. Work and energy for one degree of freedom systems.
1. Explain qualitatively the results of the following experiment. Two identical cardboard rings are hung vertically at the same height on two separ
MECH1
Problem Sheet 2
Due Friday Feb 27
Dynamics
Do all problems
1. A particle in the plane is acted upon by a force
F (x, y) = y 2 i + j.
Determine the trajectory of the particle that starts from rest at the origin.
Answer: By the second law of Newton, a
MECH1
Problem Sheet 3
Due Friday March 6
Mandatory problems: 1,3,4,5 and one out of 68.
Dynamics continued, momentum conservation, mass centre
Do all problems
1. (a) A driver going with velocity v slams his breaks. Find the distance the car travels before
MECH1
Problem Sheet 7
Due Monday May 11
Oscillations
Do all problems. Terminology: the damped harmonic oscillator equation is
x + 2 x + 2 x = 0,
if the right-hand side is not zero, it is referred to as forcing. Always assume that
.
1. (This is a revision