Hookes Law and Gravitational Acceleration Lab Report 4: Physics Answers 2022

Hookes Law and Gravitational Acceleration Lab Report 4: Physics Answers 2022

Hookes Law and Gravitational Acceleration Lab Report 4: Physics Answers 2022

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Hookes Law and Gravitational Acceleration Lab Report 4

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Contents
Abstract ……………………………………………………………………………………………………………………..2
Introduction ………………………………………………………………………………………………………………..2
Theory………………………………………………………………………………………………………………………..2
Procedure/Method ………………………………………………………………………………………………………4
Results ……………………………………………………………………………………………………………………….5
Discussion …………………………………………………………………………………………………………………..8
Conclusion ………………………………………………………………………………………………………………….8
References ………………………………………………………………………………………………………………….9
Figure 1: Object forces on an object on an incline  ………………………………………………………..2
Figure 2: Diagram of the experiment ……………………………………………………………………………….4
Figure 3: speed vs time graph …………………………………………………………………………………………6
Figure 4: distance vs time graph ……………………………………………………………………………………..6
Figure 5: representation of the incline structure ……………………………………………………………….7
1
Abstract
In this experiment we are investigating the acceleration of a ball bearing down an incline to prove if the
acceleration that is measured and the one obtained from experimenting is the same and also if the
acceleration is constant while the ball rolls down the incline, we are going to observe the relationship of
distance and time and velocity and time when acceleration is constant
Introduction
This experiment aims to investigate the acceleration of a body down a frictionless inclined slope.
This is a Galileo’s experiment, he made this experiment trying to prove that objects in free fall
will behave the same as objects on an incline . This experiment explains why a ball will roll
downhill or why a car will accelerate down a slope even when the car engine is off.
Theory
If a ball-bearing is allowed to slide down a frictionless V-shaped track inclined at an angle θ to the
horizontal, it is accelerated by the component of the acceleration due to gravity g, down the
incline:
=
When an object of mass m is placed on an inclined plane it will slide down the incline and the
greater the distance it has to slide down the faster the object becomes, object slide down an
inclined plane because of an unbalanced force. When an object is on an inclined plane that is
frictionless the object experiences two forces that act on it which are the gravitational force (Fg)
pushing the object towards the ground and the normal force(Fn) that acts perpendicular to the
surface of the object 
Figure 1: Object forces on an object on an incline 
2
Figure1 show an object on an inclined plane and the forces are not balanced because the normal is
horizontal as it should be but the force of gravity is at an angle to the surface which causes the force to
split into two forces that are of the same as the angle of incline, Mg splits into mgcosƟ, a force that
balances the normal which keeps the object on the incline and the mgsinƟ this force is not balanced by
another force so it causes an acceleration in the direction force down the incline. Because the acceleration
of this object on an inclined plane determined by the unbalanced force mgsin on a frictionless surface

= (1)
= (2)
Newton’s law:
Acceleration of an object on an incline:
=
Cancelling out mass gives us the equation of acceleration:
=
(3)
Velocity of an object
=
− 0
−0
(4)
= 0 + (5)
+
=
Average velocity (Vavg)
2
(6)
When the acceleration is constant, the average and instantaneous accelerations are equal
=
− 0
−0
= 0 +
=
Average velocity (Vavg)
+
2
1
= 0 + (7)
2
Combining the two equations (7) and (5)
1
− 0 = 0 + 2 2
(8)
The ball-bearing therefore moves with constant acceleration a, in a straight line and its displacement after
1
a time interval, t is given by: − 0 = 0 + 2 2 where V0 is the (initial) velocity at the beginning of the
1
interval . This − 0 = 0 + 2 2 equation will be used and manipulated so that it becomes the
equation of the Velocity vs. time graph.
3
The hypothesis of this experiment is that the velocity will increase when the distance is increasing and the
relationship between distance and time is that the longer the distance the ball has to travel the more time
it will need to reach the stop and because the is no external force acting on the ball the acceleration will
be constant
Procedure/Method
Apparatus

Timer
Ball-bearing
Meter ruler
Banana plugs
Connection wires
V-shape track
Stand to hold V-shape track
Marker
Before the experiment was the apparatus were checked if they were all available and all working
properly, the V-track was placed on the stand to make it an incline where the ball will roll down,
the banana plugs were connected on the timer at the start and stop, the neutral wire was
connected at the top of the V-track to ground the circuit for when the timer wires come into
contact they complete the circuit and triggers the timer, the marker is used on the V-track to with
a ruler to mark 10cm intervals from the top to the bottom of the V-track after marking place the
start wire (orange ) on the second marking from the top of the incline which is 10cm from the 1 st
mark, the 1st mark was were the ball was going to be placed and start rolling. The stop line is
placed on the 3rd marking on the track which is 10cm from the start line
Figure 2: Diagram of the experiment
4
The ball was placed on the starting line and then released and not pushed so that measurements
are constant, the timer started when the ball touched the start wire (orange) and stops when the
ball touches the stop wire (yellow), this was repeated three times on the same interval and the
measurements were recorded, the stop wire was then moved to the next line down the incline
which is 10cm from where the stop wire was previously placed this was the ball was rolled three
times and measured on the same interval, this was repeated until the stop wire (yellow) reached
the bottom, each time the stop wire is moved the timer is taken back to zero by the reset button
Results
Table 1: measured data
x (cm)
10
20
30
40
50
60
70
80
90
100

x (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t1 (ms)
56.89407
100.5504
137.3544
169.7794
199.0938
226.0512
251.1426
274.7089
296.9984
337.5143
t2 (ms)
72.36107
119.6954
152.8214
185.2464
173.6648
241.5182
266.6096
290.1759
314.7987
338.6813
t3(ms)
47.40617
91.06252
131.5562
160.2915
189.6059
216.5633
247.9806
256.5544
287.5105
328.0264
tavg (ms)
58.8871
103.7695
140.5774
171.7725
187.4549
228.0443
255.2442
273.813
299.7692
334.7407
tavg (s)
0.058887
0.103769
0.140577
0.171772
0.187455
0.228044
0.255244
0.273813
0.299769
0.334741
x/t (m/s)
1.698165
1.927349
2.134056
2.328662
2.667309
2.631068
2.742471
2.921702
3.00231
2.987387
height of stand = 25 cm
length along the floor from where the v-shaped track touches the ground to the centre of
the stand = 89 cm
(turn to next page….)
5
Figure 3: speed vs time graph
Figure 4: distance vs time graph
6
Calculations
h=0.25m

distance =0.89m
= tan−1

= tan−1
0.25
0.89
Figure 5: representation of the incline structure
= 15.689°
= sin
= 9.8 sin 15.689
= 2.650 / 2
Measured acceleration is 2.650m/S^2

1
= +

2

= . + . Is the best fit line equation
1
= 5.04
2
= 5.04 ∗ 2
= 10.08 / 2
The theoretical acceleration is 10.08m/s^2
Percentage error of the experiment
=
| − ℎ |
× 100

2.650 − 10.08
=|
| × 100
10.08
=73.715%
7
Discussion
Table 1 shows the measured results of the experiment of three tests done on a single distance
from the initial start to the finish where the stopwatch stops, the test was done 10 times to make
the experimental data more to insure more accuracy of the outcome. From the average time
column on the table, it can be noticed that when the distance between the start and the stop
increases the time also increases. On the x/t column which is velocity as the distance increases
the velocity of the ball is increasing making the ball faster.
Figure 3 is the speed vs time graph, this graph is the data for the two columns in table 1 this data
produces a positive gradient line that shows that when acceleration is constant the velocity of
the object increases as the distance it’s travelling is increasing; the gradient of this graph is the
theoretical acceleration, the gradient of the line is 5.08 from the straight-line equation obtained
from the data points, the acceleration is two times the gradient that makes the acceleration
10.08m/s^2. Figure 4 displays the relationship between distance the ball travels and the time it
takes to travel that distance it show us a positive gradient which means when the distance is
increasing the time is also increasing.
The measured acceleration which is the expected outcome is 2.65m/S^2 and the theoretical is
10.08m/s^2 the is a huge difference between the two accelerations which shows that the was an
error during the experiment, an error percentage was calculated and a 73% error was to the
outcome which proves that the data has an error. Friction on the experiment was neglected but
because this is a real-life experiment simulation is always present and it might have played a
bigger role in the results, on some of the tests the demonstrator might have pushed the ball
without noticing and that also might have contributed to the error.
Conclusion
When a ball was placed on an incline of a certain height the object will accelerate downward and
when the distance of the incline where to ball will be rolling on is increased the velocity of the
ball also increases but the acceleration stays constant, the acceleration of an object on an inclined
plane will stay constant as it rolls down the incline while the velocity increases which proves the
hypothesis correct. The aim of this experiment was reached as the relationship between distance
and time was proved and the acceleration of a object on an incline is constant always if no
external force acts on it .On this experiment we learned that an object that is on a frictionless
incline plane without any external forces acting on it will behave as a free falling object
8
References
 Inclined-Planes, retrieved 13 June 2021 from
https://www.physicsclassroom.com/class/vectors/Lesson-3/Inclined-Planes.
 Inclined-plane, retrieved 12 June 2021 from
 Galileo’s Inclined Plane Experiment, retrieved 12 June 2021 from
https://www.maplesoft.com/support/help/maple/view.aspx?path=MathApps%2FGalileosInclinedPlane
Experiment
9
EXPERIMENT 2
Hooke’s law
Aim
To determine a spring-constant (k). This will be done through investigating Hookes
law by attaching weights to a spring and observing how the displacement of the spring
changes with the number of weights attached to the spring. Each weight has a mass of
100 grams. The force diagram is shown in Fig. 2 below.
Figure 2: A diagram that shows the relationship between the restoring force and the
weight of the mass m. Taken from https://en.wikipedia.org/wiki/Hooke’s law.
Introduction and Theory
Hooke’s law states that the extension of a spring is directly proportional to the load
applied to the spring. In mathematical form, the Hooke’s law is expressed as:
F = −kx
(5)
Where F is the restoring force that tends to push the spring back to its equilibrium
point, k is the spring constant, and x is the displacement (or extension of the spring).
The negative sign in the above equation indicates that the restoring force always acts
in a direction opposite that of the displacement.
8
Method
You are provided with five copper weights, each measuring 100g. You are also provided
with a metre ruler, a spring and a wire hook.
Attach the end of the spring to the stand, and fix a wire hook to the other end of
the spring. The purpose of the wire hook is to act as a platform on which to attach
copper weights. By attaching several weights to the wire hook you increase the weight
of the load attached to the spring, and thus alter the restoring force.
Mark the rest position of the wire hook attached to the spring. This is your zero
displacement (extension) position. Include this position in the table of measured values. You are provided with five copper weights, attached the first one to the metal
hook (which is attached to the spring), and note the displacement (you can mark with
a pencil and use a ruler afterwards to measure the distance from zero displacement).
Keep on adding the copper weights until all of them are attached to the spring, each
time mark their respective displacement. You should by now have six measurement of
the displacement (starting from zero), and six readings of the cumulative weight of the
copper weights. Tabulate your measurement (taking note of significant figures and the
rules we discussed last week) as shown below in table 2:
Displacement Displacement Mass suspended
(cm)
(m)
(kg)
Weight
Restoring
(N)
Force (N)
Table 2: A table of measured displacement and the restoring force on the spring.
You are provided with a MATLAB program called fit.m (see Appendix E.1). Your
practical lecturer will explain to you how to use the program to fit a straight line to
the data above (restoring force vs. displacement). Compare the fitting function of the
9
form:
F(x) = mx + c
(6)
With Eq. 5, and use the slope obtained from the fitting function (and the error in the
slope) to write down the spring constant. Please remember that it is always important
to write down the units of each measurement.
• Is your data nicely described by Eq. 5 above? If not, what is the reason for the
discrepancy.
• What is the value of the spring constant and its error, use the standard scientific
way of presenting a measurement and its error.
10
EXPERIMENT 4
Measuring gravitational acceleration from the fall of
a ball-bearing
Aim
The aim of this experiment is to measure the gravitaional constant (g) by studying the
motion of a ball-bearing falling vertically under gravity.
Apparatus
You are provided with the following apparatus:
• A digital timer
• Ball-bearing
• A stand
• A release mechanism mounted on the stand
18
Figure 5: The experimental setup Pictures (courtesy of D. Nhlapo).
Figure 6: A pictures of a release mechanism which shows a release knob and clamping
nut (courtesy of D. Nhlapo).
19
Introduction and Theory
The motion of an object falling under gravity is described by Newton’s second law: F
= ma. The height from which the falling object is released is related to the time it takes
to fall by:
1
h = v0 t + gt2
2
(26)
where v0 is the initial speed of the object and g is the gravitational constant. If the
object falls from rest, when v0 is zero and equation 26 becomes:
1
h = gt2
2
(27)
Therefore, the bigger the height, the longer is the time an object will take to fall to
the ground. Equation 27 can be used to determine the value of g by investigating the
relationship between height (h) and time (t). According to equation 27, the height is a
quadratic function of time. We therefore cannot use measurements of h to get g. But
if we plot h vs. t2 , then we can get a sraight line, the slope of which gives half of the
value of g. This is the method we will use to measure g.
Method
The stand has a meter rule mounted on it, this is used to measure distance of fall between the release mechanism and the landing pad. The meter rule also has two markers
to make it easy to measure the position of the landing pad and release mechanism.
Before you take the measurements, ensure that the display button on the digital
timer is set to milliseconds ms, and the function button is set to timer. Ask the lab
technician or facilitator for assistance with this.
Clamp the ball-bearing by pushing the release knob and tighten the clamping nut
(screw next to the release knob) on the release cable. This will hold the ball-bearing in
place.
20
Press the stop button, then reset button and start button on the digital
timer (in that order). The green light above the start button will turn on.
Raise the landing pad by pulling it up until it cannot move anymore. You are now
Note the distance between the release mechanism and the landing pad. Take care
of the significant figures. Release the clamping nut by unscrewing it slowly, the ballbearing will fall, starting the counting of the timer. When the ball-bearing hits the
landing pad, the counting on the timer will stop. Vary the distance between release
mechanism and landing pad. Make a table like the one in table 4 below.
h (m) t (s) t2 (s2 )
Table 4: A table of measurements .
Perform the following after collecting the data:
• Write a MATLAB program to plot height h versus time t. Is the plot linear
with units included.
• Use the MATLAB program, fit.m, that is provided to you in Appendix E.1 to fit
the data in table 4 (as discussed in the introduction and theory section above) to
estimate the value of g. What is the error in your measurement? is the h
vs. t2 plot linear? it not, why?
Reference
• Halliday, D., Resnick, R., Walker, J., Fundamentals of Physics, 9th Edition
21
height (cm) time (ms)
48.2
323.08
47.5
308.33
43
292.48
39.9
282.47
32.7
253.62
0
0

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