FACULTY OF MECHANICAL & MANUFACTURING ENGINEERING
BDA20103 DYNAMICS
SESSION 2021/2022
MINI GROUP PROJECT
TOYS CAR ROLLING DOWN AN INCLINED PLANE
GROUP
GROUP MEMBERS
GROUP
LECTURER
SUBMISSION DATE
3
1. MOHAMED AIMAN ARIF BIN MOHAMED NAZRI
(CD210061)
2. MUHAMMAD AFFENDY BIN ZAINAL ABIDIN
(CD210086)
3. MOHAMMAD DANIEL FAIZ BIN MOHD
NORRISHAM (CD210053)
4. MOHD SYAHIR BIN ABD RAZAK (CD210011)
5. MUHAMAD ARIFF LUKMAN BIN MOHD SUIB
(AD200077)
6. NOHD DIMILBIN ABDULLAH (CD200030)
3
DR. NORADILA BINTI ABDUL LATIF
26/6/2022
TABLE OF CONTENTS
CHAPTER 1 – INTRODUCTION ....................................................................................... 1
CHAPTER 2 – CASE STUDY ............................................................................................. 2
CHAPTER 3 – CALCULATION AND ANALYSIS .................................................................. 6
CHAPTER 4 – DISCUSSION ............................................................................................ 10
CHAPTER 5 – CONCLUSION AND RECOMMENDATION ......................................... 11
REFERENCES ................................................................................................................... 12
APPENDICES .................................................................................................................... 13
CHAPTER 1: INTRODUCTION
1.1 Introduction
What is dynamic meaning? Dynamics is a branch of physical science and a subset of
mechanics that studies the motion of material objects in relation to the physical factors that
influence them: force, mass, momentum, and energy. There are numerous things in our
environment that make use of dynamic calculation knowledge. We will be able to understand the
principle of rigid body mechanics and kinematics by mastering the broad course of dynamics.
Furthermore, dynamics includes the calculation of the motion of any object, whether in linear or
curvilinear motion, such as calculating the component velocity of cars after impact.
We opted to undertake a case study for this project that demonstrates the study of the
kinematics of a rigid body with a ball rolling down an inclined plane at an angle. This project is
modelled by Galileo Galilei's experiment. Isaac Newton developed the Law of Inertia as a result
of this. In this scenario, we wish to measure the ball's acceleration as a function of angle and
compare the experimental results to the theoretically predicted results.
1.2 Objectives
The purpose of this collaborative effort was to accomplish numerous goals, including:
i.
To find the final velocity of the toys lorry move down the incline using kinematics and
conservation of energy and consider the lorry move without slipping.
ii.
To find the acceleration of the lorry move the incline using the kinematics of the rigid
body.
iii.
To find the difference in acceleration and velocity between the three different sizes and
weight of lorry
CHAPTER 2 – CASE STUDY
2.1 Case Study Description
Every child's fascination with the world of physics begins with a basic situation that
they simply cannot comprehend. For example, they might notice a small pebble rolling down
a steep road and ask why it is rolling away. Humans' curious curiosity, as well as the neverending wonders of physics, enable us better understand the world. Many things contribute to
the motion of a simple occurrence like a rock rolling down a hill. As we delve more into the
scenario, we discover that the Newton's Second Law of Motion is in effect, with the force, F
= mass x acceleration. We can virtually precisely compute the rock's acceleration as it rolls
with correct analysis. Is it true that various masses slide down a slope with differing
kinematic properties? This mini project focuses on this part of the case study, in which we
conduct experiments to investigate the differences in the kinematics of stiff bodies of various
masses. We can adequately grasp how an object's mass impacts its kinematic qualities such
as acceleration and velocity as it rolls down an inclined plane by keeping all other variables
constant.
Figure 1 : shows an illustration of rock with different masses rolling down a hill.
2.2 Methodology
In order to better understand how the mass of an object affects its kinematics as
it rolls down an inclined plane, our group came up with a simple experiment. We set
up an incline with a angle of incline at 30°,40°,50° and fixed distance of 1m. We then
chose 3 car that have different masses, which are 0kg, 100kg and 150kg
respectively. The material of the ball and the surface of the incline is kept constant to
make sure other factors such as friction and air resistance do not affect the readings.
First the car with 0kg mass is set up at the top of the incline. Without accidentally
applying any force to the car, the car is let to roll from the top of the incline to the
bottom. The experiment is then repeated with the balls of mass 100kg and 150kg. At
the end of the experiment, the normal force acting on the car, the acceleration and
velocity of the car, the kinetic energy of the balls and the impulse that the car contain
are calculated.
lOMoARcPSD|133 520 57
Figure 2 shows an example set up of the experiment.
To find the normal force, the equation used is Newton’s Second Law of Motion:
F=ma
where:
F = the force that the car exerts onto the incline
m = the mass of the car
a = gravitational acceleration exerted onto the car due to
gravity
The acceleration of the car can also be calculated using the same equation. The velocity of the
ball, however, can be calculated using one of the equations of motion:
2
= 2 + 2as
where:
= the final velocity of the car,
= the initial velocity of the car at the top of the ramp (0 m/s) , a = the
acceleration of the car, s = the distance travelled by the ball/the length of the
inclined plane (1m).
lOMoARcPSD|133 520 57
The kinetic energy of the car was calculated using the formula:
KE= mv2
where:
m = the mass of the car, and
= the final velocity of the car.
2
Downloaded by aaa bbb (ndsfn1@vintomaper.com)
lOMoARcPSD|133 520 57
2.3 Experiment Results
Based on the figure above, the case study is carried out based on theoretical analysis
since we are not able to make use of the apparatus to conduct the case study physically. Three
balls of different masses are placed at the beginning of the plane, s1 and released until it reached
the end of the plane, s2. The plane distance, s is 1meter long. Through this we apply the
kinematics of rigid body and calculate the velocity and acceleration accordingly.
As mention, the result are based on theoretical analysis under the kinematics of rigid
body. the results can see at calculation part
Calculation
The load carry by the lorry and angle of the inclined plane 𝜃 = 30°, 40°, 50° and wood
surface and distance is maintained constant.
Normal force
To calculate normal force, use Newton’s second law of motion, 𝑓 = 𝑚𝑎; (↑ +) ∑ 𝐹 = 𝑚𝑎
where there is no acceleration, 𝑎 and 𝜃 = 30°, 40°, 50°.
𝜃 = 30°
i.
𝑚 = 0.05𝑘𝑔
𝑁 − (0.05)(9.81) cos 30° = (0.05)(0)
𝑁 = 0.42𝑁
ii.
𝑚 = 0.1𝑘𝑔
𝑁 − (0.1)(9.81) cos 30° = (0.1)(0)
𝑁 = 0.85𝑁
iii.
𝑚 = 0.15𝑘𝑔
𝑁 − (0.15)(9.81) cos 30° = (0.15)(0)
𝑁 = 1.27𝑁
𝜃 = 40°
i.
𝑚 = 0.05𝑘𝑔
𝑁 − (0.05)(9.81) cos 40° = (0.05)(0)
𝑁 = 0.37𝑁
ii.
𝑚 = 0.1𝑘𝑔
𝑁 − (0.1)(9.81) cos 40° = (0.1)(0)
𝑁 = 0.75𝑁
iii.
𝑚 = 0.15𝑘𝑔
𝑁 − (0.15)(9.81) cos 40° = (0.15)(0)
𝑁 = 1.13𝑁
𝜃 = 50°
i.
𝑚 = 0.05𝑘𝑔
𝑁 − (0.05)(9.81) cos 50° = (0.05)(0)
𝑁 = 0.31𝑁
ii.
𝑚 = 0.1𝑘𝑔
𝑁 − (0.1)(9.81) cos 50° = (0.1)(0)
𝑁 = 0.63𝑁
iii.
𝑚 = 0.15𝑘𝑔
𝑁 − (0.15)(9.81) cos 50° = (0.15)(0)
𝑁 = 0.94𝑁
Acceleration
To calculate acceleration, use Newton’s second law of motion; 𝐹 = 𝑚𝑎; (→ +) ∑ 𝐹𝑥 =
𝑚𝑎𝑥 , given coefficient of kinetics friction (𝜇𝑘) is 0.4.
𝜃 = 30°
i.
𝑚 = 0.05𝑘𝑔
(0.05) sin 30° − 0.4(0.42) = (0.05)𝑎𝑥
𝑎𝑥 = 2.86 𝑚⁄𝑠 2
ii.
𝑚 = 0.1𝑘𝑔
(0.1) sin 30° − 0.4(0.85) = (0.1)𝑎𝑥
iii.
𝑎𝑥 = 2.90𝑚⁄𝑠 2
𝑚 = 0.15𝑘𝑔
(0.15) sin 30° − 0.4(1.27) = (0.15)𝑎𝑥
𝑎𝑥 = 2.89𝑚⁄𝑠 2
𝜃 = 40°
i.
𝑚 = 0.05𝑘𝑔
(0.05) sin 40° − 0.4(0.37) = (0.05)𝑎𝑥
𝑎𝑥 = 2.32 𝑚⁄𝑠 2
ii.
𝑚 = 0.1𝑘𝑔
(0.1) sin 40° − 0.4(0.75) = (0.1)𝑎0
𝑎0 = 2.36𝑚⁄𝑠 2
iii.
𝑚 = 0.15𝑘𝑔
(0.15) sin 40° − 0.4(1.13) = (0.15)𝑎0
𝑎0 = 2.37𝑚⁄𝑠 2
𝜃 = 50°
i.
𝑚 = 0.05𝑘𝑔
(0.05) sin 50° − 0.4(0.31) = (0.05)𝑎0
𝑎0 = 1.71 𝑚⁄𝑠 2
ii.
𝑚 = 0.1𝑘𝑔
(0.1) sin 50° − 0.4(0.63) = (0.1)𝑎0
𝑎0 = 1.75𝑚⁄𝑠 2
iii.
𝑚 = 0.15𝑘𝑔
(0.15) sin 50° − 0.4(0.94) = (0.15)𝑎0
𝑎0 = 1.74𝑚⁄𝑠 2
3.1.3 Velocity
The velocity can be determined by using an equation of motion with constant acceleration
𝑉 2 = 𝑉𝑜 + 2𝑎𝑠 , where s is a constant throughout 100cm, and an object is start from rest and
Vo is 0 m/s.
For 30°
i.
m1 = 0.05 kg
v² = 0² + 2 (2.86) (1.0)
v² = 5.72
v = 2.392 m/s
ii.
m2 = 0.1 kg
v² = 0² + 2 (2.90) (1.0)
v² = 5.80
v = 2.4083 m/s
iii.
m3 = 0.15 kg
v² = 0² + 2 (2.89) (1.0)
v² = 5.78
v = 2.404 m/s
For 40°
i. m1 = 0.05 kg
v² = 0² + 2 (2.32) (1.0)
v² = 4.64
v = 2.15 m/s
ii. m2 = 0.1 kg
v² = 0² + 2 (2.36) (1.0)
v² = 4.72
v = 2.17 m/s
iii. m3 = 0.15 kg
v² = 0² + 2 (2.37) (1.0)
v² = 4.74
v = 2.18 m/s
For 50°
I.
m1 = 0.05 kg
v² = 0² + 2 (1.71) (1.0)
v² = 3.42
v = 1.85 m/s
II.
m2 = 0.1 kg
v² = 0² + 2 (1.75) (1.0)
v² = 3.50
v = 1.88 m/s
III.
m3 = 0.15 kg
v² = 0² + 2 (1.74) (1.0)
v² = 3.48
v = 1.86 m/s
3.1.4 Kinetic Energy
The kinetic energy was calculated by using this formula:
For 30°
𝑲𝑬 =
𝟏
𝒎𝒗²
𝟐
1. m1 = 0.05 kg
1
KE = (0.05) (2.392)²
2
KE = 0.1430 J
2. m2 = 0.1 kg
1
KE = (0.1) (2.4083)²
2
KE = 0.2899 J
3. m3 = 0.15 kg
1
KE = (0.15) (2.404)²
2
KE = 0.4334 J
For 40 °
𝟏
KE= mv²
𝟐
1. m1 = 0.05 kg
1
KE = (0.05) (2.15)²
2
KE = 0.1155 J
2. m2 = 0.1 kg
1
KE = (0.1) (2.17)²
2
KE = 0.2354 J
3. m3 0.15 kg
1
KE = (0.15) (2.18)²
2
KE = 0.3564 J
For 50°
𝟏
KE = mv²
𝟐
1. m1 = 0.05kg
1
KE = (0.05) (1.85)²
2
KE = 0.0856 J
2. m2 = 0.1 kg
1
KE = (0.1) (1.88)²
2
KE = 0.1767 J
3. m3 = 0.15kg
1
KE = (0.15) (1.86)²
2
KE 0.2595 J
3.2 Analysis
i.
Normal Force
When the mass of an object increases and the acceleration of y remains constant,
the normal force acting perpendicular to the surface increases.
ii.
Acceleration and velocity
Surprisingly, even though the object's mass changes, the accelerations and
velocities remain constant. As a result, the object mass has no effect on the
values of velocity and acceleration.
iii.
Kinetic energy
As a result, the greater the mass of an object, the more kinetic energy can be
stored within it due to constant accelerations and velocities.
iv.
Impulse
The total impulse that occurs in a time period, like kinetic energy, can be
affected by the mass of the object. As the mass of the object increases, the
incline becomes steeper and the surface smoother, resulting in a higher value of
impulse.
CHAPTER 4 – DISCUSSION
In this case study, our group has decided to conduct a study on the kinematics of rigid
bodies in the case of trucks with various masses moving down an inclined plane with
constant angles. The primary justification for this specific case study is the straightforward
yet concise information it provides regarding normal force, acceleration, velocity, and kinetic
energy. It is a comprehensive experiment that is comparatively simple to study.
The primary purpose of this case study, to calculate the acceleration of a rigid body
(the lorry) moving down an inclined plane at a specific angle using the kinematics of rigid
bodies formula, is what this experiment aims to achieve first. Given that friction is so
minimal, the shape and size of the lorry makes it easy for us to assess energy conservation.
Finding the differences in acceleration and velocity between three different loads on a lorry is
the final goal.
As mention, we had set a few parameters for this case study. Firstly, the inclination of
the plane is set at angle 30°, 40° and 50°. The mass of the lorry is set to 0.05kg, 0.1kg and
0.15kg and we considered the plane to be frictionless in order to find the magnitude of
acceleration and velocity of the lorry. From the calculation, we get the magnitude of
acceleration at angles 30°, 40° and 50° are 2.9m/s², 2.3m/s² and 1.7m/s². Besides, the
velocities at angle 30°, 40° and 50° are 2.4m/s, 2.1m/s and 1.8m/s.
Through this case study, we're learn how to use a rigid body's kinematics and how
they affect the lorry. The recommendation for this study is to use lorry made out of metal or
any material with less frictional effect.
CHAPTER 5 – CONCLUSION AND RECOMMENDATION
We can infer from the case study we conducted and the computations we examined
that changing an object's mass has no impact on its velocity and acceleration. However, for
the kinetic energy we can conclude that the greater the mass of the object, the higher the
kinetic energy will be stored inside the object even during constant accelerations and
velocities. Furthermore, the mass of an object can also have an impact on the overall impulse
that occurs over a given time period because the value of mass rises, the inclination gets
steeper and the surface also will become smoother. Therefore, an object with a larger mass
will have a higher impulse value than an object with a lower mass.
By choosing three distinct starting points for the object's descent down the inclined
plane, we may compare the results and come up with a proposal. As a result, we can observe
the difference in acceleration and velocity at each initial distance point.
lOMoARcPSD|133 520 57
REFERENCES
1) Hibbeler, R. (2015). Engineering Mechanics: Dynamics (14th ed.). Pearson.
2) Palanichamy, M. S., Nagan, S., & Elango, P. (1998). Engineering Mechanics. McGrawHill Education.
3) Department of Physics, University of Illinois at Urbana-Champaign. (2007, October
22).
Balls
Rolling
Down
a
Ramp.
AsktheVan.
https://van.physics.illinois.edu/qa/listing.php?id=183&t=balls-rolling-down-the-ramp
4) Catharine H. Colwell. (2021). PhysicsLAB: Galileo Ramps. PhysicsLab.
http://dev.physicslab.org/Document.aspx?
doctype=2&filename=Kinematics_GalileoRamps.xml
5) GALILEO’S KINEMATICS. (2010). Sci122 Lab - Kinematics. Science 122
Laboratory.
https://www.honolulu.hawaii.edu/instruct/natsci/science/brill/sci122/SciLab/L6/kinela
b.html
6) Daniel M. (2017, April 13). Ball Rolling Down an Incline - IB Physics. YouTube.
https://www.youtube.com/watch?v=1hAjj1mgm-4