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Mechanics Lab (Civil Engineers)
Name (please print): _________________________
Tutor (please print): _________________________
Lab group: ________________________________
Date of lab: ________________________________
Experiments
In the session you will be divided into groups and perform four separate experiments:
(1)
air-track collisions;
(2)
static and dynamic friction;
(3)
flywheel experiment;
(4)
mass-spring oscillations.
Write-Up and Submission
●
Experimental data should be recorded during the laboratory session. Subsequent
analysis can be completed during the lab or later. Results and analysis should be
submitted on this worksheet.
●
Measurements are taken in groups, but analysis must be your own individual work.
Students found colluding or copying will be subject to disciplinary action.
●
Submission is to the appropriate coursework box by 11 pm on the same day of the
week, one week after you take the lab. Work will be returned when everybody’s
report has been marked.
●
Work submitted late or to the wrong coursework box will not be marked. Poor
presentation (including excessive unnecessary material) will also lose marks.
●
This report constitutes 10% of the total marks for the Mechanics unit.
Mechanics (Civil)
Labs - 1
David Apsley
1. Air Track Collisions
The air track provides a nearly frictionless environment for experiments on objects moving in
a straight line. You will use two “gliders”, one initially stationary near the middle of the track
and one set in motion by an elastic system. There is also a measuring system, which uses
optical gates to determine the speed of the gliders.
optical gates
electromagnet
moving
glider (1)
stationary
glider (2)
Method
The optical gates should be either side of the starting point of the stationary glider. The other
glider is held against an elastic end stop by an electromagnet. When the magnet is switched
off, its glider is propelled forward by the elastic. Its initial velocity can be determined as it
passes the optical gate. After the collision, the final velocities of the two gliders can be
determined as they pass one or other of the optical gates. You may need to adjust the position
of the gates so that only one glider is passing a gate at a time, and you may need to catch one
of the gliders.
The mass of each glider is roughly 200 g, but you should measure these accurately. There are
some additional masses that can be added to one or both of the gliders. Do the experiment
with three combinations of masses. Write m1 and m2 for the masses of the two gliders, u1 and
u2 for the initial velocity of each glider, and v1 and v2 for the signed velocities (negative if in
the opposite direction to the initial motion).
Mechanics (Civil)
Labs - 2
David Apsley
Results Tables
Record your measurements in the following table.
m1 (kg)
m2 (kg)
u1 (m s–1)
u2 (m s–1)
v1 (m s–1)
v2 (m s–1)
Analyse your results in the following table.
Initial momentum Final momentum
Speed of
Speed of
–1
–1
–1
(kg m s )
(kg m s )
approach (m s ) separation (m s–1)
Coefficient of
restitution, e
0
0
0
Questions
1.
For each experiment, calculate the total momentum before (m1u1 + m2u2) and the total
momentum after (m1v1 + m2v2). What result do you anticipate obtaining?
2.
For each experiment, calculate speed of approach, speed of separation and thus the
coefficient of restitution, inserting values in the table.
speed of separation v2  v1
e

speed of approach u1  u 2
What would a value e = 1 mean?
What would a value e = 0 mean?
Suggest sources of error in the experiment.
Mechanics (Civil)
Labs - 3
David Apsley
2. Static and Dynamic Friction

A mass m is initially resting on an inclined slope of angle θ. As the angle is increased the
component of weight directed down the slope increases and eventually exceeds the static
friction between the mass and the slope. This can be used to determine the coefficient of
static friction (μs) from
μ s  tan θ s
where θs is the angle of incipient motion.
If the mass is given an initial push to get it moving, it will be subject to kinetic friction.
Friction and downslope weight component will be in equilibrium (and hence the mass will
just keep moving) when
μ k  tan θ k
For 3 different objects:
●
measure the angle θs at which the stationary mass just starts to slide, and hence
deduce the coefficient of static friction, μs;
●
measure the angle θk at which a moving mass just stops sliding, and hence deduce the
coefficient of kinetic friction, μk;
●
calculate the ratio μk/μs.
Repeat this for one of the objects with an additional mass on top to increase the normal
reaction.
Mechanics (Civil)
Labs - 4
David Apsley
Object
Angle of
incipient sliding,
θs
Angle of
incipient stopping,
θk
Coefficient of
static friction,
μs
Coefficient of
kinetic friction,
μk
1
2
3
Object + mass
By considering the balance of forces, prove analytically the result μ  tan θ (applying to both
static and kinetic friction).
Do you expect the ratio μk/μs to be the same for all materials? On what might it depend?
Mechanics (Civil)
Labs - 5
David Apsley
I
3. Flywheel Experiment
r
The moment of inertia I plays a similar role in rotational
mechanics to that which mass does in linear mechanics.
In this experiment a falling weight is used to turn a flywheel. The
time and distance of fall are measured. These may be used to
calculate the acceleration of the falling body and thence the
moment of inertia of the flywheel.
a
m
mg
Method
A weight mg is attached to a string which is initially wrapped around the shaft of a flywheel.
The shaft has radius r = 0.011 m (diameter 22 mm). The mass is allowed to fall a distance h
and the time taken, t, measured with a stopwatch. The acceleration a is then derived from
h  12 at 2
In an ideal system, the linear acceleration, a, of the mass is related to the moment of inertia of
the flywheel, I, and the acceleration due to gravity, g (=9.81 m s–2), by
g
a
I
1
mr 2
(You will be able to derive this once we have covered rotational dynamics.) Since I is much
larger than mr2 here, the “+1” in the denominator can be neglected, so that
mgr 2
a
I
Rearrange this to determine the moment of inertia I.
Find the final velocity v of the mass from a constant-acceleration formula and the final
angular velocity ω of the flywheel by rearranging
v  rω
Check your calculations by comparing the loss of potential energy (mgh) with the gain in
kinetic energy of the falling mass ( 12 mv 2 ) and flywheel ( 12 Iω 2 ).
Repeat for a different falling weight.
Mechanics (Civil)
Labs - 6
David Apsley
Results
Record your measurements in the following table.
mg
h
t
(N)
(m)
(s)
Analyse your results in the following table.
Acceleration Moment of
Final
Angular
inertia
velocity
velocity
a
I
v
ω
(m s–2)
(kg m2)
(m s–1)
(rad s–1)
Loss of PE
mgh
(J)
KE of
mass
2
1
2 mv
(J)
KE of
flywheel
2
1
2 Iω
(J)
Comment on the energy changes in the experiments – which types are involved; which
increase and which decrease – and the relative sizes of the kinetic energy of the mass and the
flywheel.
Suggest sources of error in the experiment and, if possible, ways of eliminating them.
Mechanics (Civil)
Labs - 7
David Apsley
4. Oscillations
A mass m is hung from one or two equivalent springs in each of the configurations shown.
The period of oscillation T is compared with that predicted theoretically:
m
T  2π
k eff
where keff is the equivalent effective stiffness (the stiffness of a single spring that would have
the same effect on a suspended load as a multi-spring configuration).
single spring
parallel springs
k
k
m
k
series springs
k
m
k
m
Method
●
●
●
By measuring the extension of a spring for a given load, measure the stiffness k of a
single spring. The tension F in the spring and its extension x are related by F = kx.
For each of the configurations shown, measure the period of oscillation by timing a
small number of oscillations. Try at least two masses for each combination.
Compare measured periods with those predicted from the mass m and effective
stiffnesses keff.
Single spring stiffness – use the average of several readings
Force
F (N)
Extension
x (m)
Stiffness
k (N m–1)
Average:
Mechanics (Civil)
Labs - 8
David Apsley
Configuration:
Mass
m (kg)
Measured period
T (s)
Effective stiffness
keff (N m–1)
Predicted period
Tpred (s)
Single spring
Parallel springs
Series springs
Show how you deduce equivalent effective stiffnesses keff for:
– parallel springs:
– series springs:
Complete the table for measured and predicted oscillation periods.
Explain physically which combination you would expect to have the shortest, and which the
longest, oscillation period and why.
Mechanics (Civil)
Labs - 9
David Apsley