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Using Everyday Examples in Engineering (E3)
Introduction to Civil Engineering: Engineering Elements Embedded in a Child’s Toy
Indranil Goswami
Morgan State University
Photo used with permission of International Playthings LLC
Objective of the game “Frog Hoppers”
The game Frog Hoppers is a simple child’s (3+) toy whose objective is to use elastic spring-like
element (the frog’s tail) to make the plastic frog jump into a plastic bucket, which also serves as
the container. The frogs come in 4 different colors, so that 4 individuals or teams can make it a
competitive game.
The objective of this exercise is to discover elements of engineering in this toy. Believe it or not,
there are about 4 to 5 groups of engineering concepts embedded in the design of this simple toy.
We are going to review the following concepts through our learning modules:
1. Measurements – use of precise equipment (Vernier caliper) for measurement
2. Kinematics – projectile motion
3. Structural Mechanics – bending stiffness of beams
4. Work and Energy – conversion of elastic (spring) energy into kinetic energy
5. Dynamic Coupling – between support and structure
6. Strength of Materials – fatigue strength
7. Aerodynamics – drag resistance
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of NSF.
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Learning Module 1: Measurements
Use a Vernier caliper to measure the following dimensions. See figure 4.1 to identify these
variables.
1. Length of the frog tail, L (mm)
2. Width of the frog tail, b (mm)
3. Thickness of the frog tail, t (mm)
4. Mass of the frog, m (mg)
LEARNING OBJECTIVE 1
List the measured variables in table 1 below:
Table 1: Summary of physical properties
Variable Name Value
Units
L
b
t
m
Learning Module 2: Kinematics
Once a ‘particle’ is released in the gravitational field with a velocity Vo as shown in figure 2.1
below, the theoretical path of travel is parabolic. If the origin is located at the point of launch, the
equation of the parabola is
Equation 2.1
Vo
H
α
R
Figure 2.1: Projectile Motion of a Particle in a Gravitational Field
Since the launch angle α and launch velocity Vo are initial values (constant), this equation
describes a parabola.
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The theoretical range (R) of the projectile can be obtained by setting y = 0 in the equation
Equation 2.2
The theoretical maximum height (H) of the projectile can be obtained by setting dy/dx = 0 in the
equation
Equation 2.3
LEARNING OBJECTIVE 2
Since it is difficult to measure the launch velocity Vo, use the ratio of equations 2.3 and 2.2 as
shown below to estimate the launch angle α
Equation 2.4
Measure H and R for each launch and estimate launch angle α using equation 2.4
Since the measurement of R is more reliable than H, use equation 2.2 to calculate the launch
velocity Vo using the measured R and the estimated α
For launches 1 and 2, push the tail-spring all the way down to the floor. Since this gap can be
pre-measured, this gap is equal to Δ. For launches 3 and 4, place a dime on the floor and push the
tail-spring to the top of the dime. Since the thickness of the dime can be pre-measured, the
difference between the original gap and the thickness of the dime is equal to Δ. For launch 5, use
a quarter in a similar way. Summarize your results in table 2 below.
Table 2: Summary of kinematic data for launches 1-5
Launch
H
R
Δ
No.
(mm)
(mm) (mm)
1
2
3
4
5
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Learning Module 3: Structural Mechanics – Bending Stiffness of Beams
Figure 3.1 shows a cantilever beam (fixed at one end and free at the other end) loaded with a
transverse point load at the free end. This is how we are modeling the frog’s tail. Also, see figure
4.1
P
Δ
Figure 3.1
Bending of a Cantilever Beam under Tip Load
Elastic theory can be used to demonstrate that the (maximum) tip deflection of a cantilever beam
is given by
Equation 3.1
where E is the modulus of elasticity of the beam material. Typically, the modulus of elasticity or
Young’s modulus is determined from a tensile test. The initial slope of the stress-strain diagram
from such a test is the modulus of elasticity E.
LEARNING OBJECTIVE 3
A steel (E = 200GPa) ruler which has a thickness of 1 mm and width of 25 mm is rigidly
clamped at one end and pushed with a steady, static force by placing a mass of 1 kg on the end in
a manner similar to the figure above. Calculate the static deflection of the end (mm).
Calculate the stiffness of the beam described above. Note that the stiffness of the beam is a
property of its material and section geometry, and is independent of the load.
Learning Module 4: Application of Work and Energy Principles
The frog is launched into motion by pushing down on its tail, thereby storing elastic energy.
Upon release, this energy is converted into kinetic energy, resulting in the frog attaining a launch
velocity Vo. If it is assumed that the potential energy of the frog at the instant of launch is zero
(neglecting the slight depression below the datum), then equating the spring elastic energy to the
frog’s kinetic energy, we have
Equation 4.1
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The spring stiffness k is the elastic stiffness of a cantilever beam given by
Equation 4.2
L
b
Δ
Figure 4.1
t
Dimensions of Various Elements of the Elastic Spring (Tail)
Equation 4.3
The mass of the frog (m) is to be measured precisely, Vo has been previously estimated, L, b and
t are the length, width and thickness of the tail spring – all of which can be measured quite
precisely using the Vernier caliper.
Thus, using the measured parameters m, L, b, t and Δ and the calculated parameter Vo, we can
estimate the elastic modulus E for the plastic material.
LEARNING OBJECTIVE 4
For each of the launches tabulated in module 2, the vertical deflection of the tail-spring is
controlled (see next page) and recorded in the last column of table 3.
For each launch, the modulus of elasticity is calculated using the following equation
Equation 4.4
Rewrite the data from table 1 into the first 5 columns of table 3 below. For launches 1 and 2,
push the tail-spring all the way down to the floor. Since this gap can be pre-measured, this gap is
equal to Δ. For launches 3 and 4, place a dime on the floor and push the tail-spring to the top of
the dime. Since the thickness of the dime can be pre-measured, the difference between the
original gap and the thickness of the dime is equal to Δ. For launch 5, use a quarter in a similar
way.
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Table 3
Launch H
No.
(mm)
R
(mm)
α
(degrees)
Vo
(mm/sec)
Δ
(mm)
E
(GPa)
Learning Module 5: Dynamic coupling between support and structure
The lid of the plastic bucket doubles as a launching surface. The added elasticity of the lid serves
to increase the range (R) of the frog. By using two separate launch positions for the frog, make
multiple measurements of the range.
Procedure:
1. Mark two positions on the launch lid – one exactly at the center of the lid and another
from a rigid surface such as a tabletop
2. Make 5 launches from each position. In each case measure the range and compute the
average range. Also compute the standard deviation of the measurements and comment
on the variation of these measurements.
3. Compare the range from the flexible surface to the range from the rigid surface.
Comment.
Learning Module 6: Strength of Materials – Fatigue Strength
You must appreciate that in the choice of material for the frog, material behavior under repeated
loading must play an important role. How a material responds to the repeated cyclic loading and
unloading as opposed to supporting a sustained load for a long time is key in determining the
durability of the material and the long-term quality of the toy. Research online using the
keywords ‘fatigue strength’ and write a definition for it.
Learning Module 7: Aerodynamics – Drag Resistance
The kinematic equations presented in section 2 and then used in this paper are based on particle
kinematics, i.e. for an object that can be modeled as a particle (occupying no space). For such an
object, there would be no aerodynamic drag. However, our plastic frog does have significant
drag potential and therefore, particle kinematics is not strictly valid. As a result, our conclusions
do have a certain degree of error.
In designing experiments, it is always important to understand the various sources of systematic
error that are embedded in the methodology (in addition to random human error that can always
occur, even in perfectly designed experiments). The presence of such errors, produced as a result
of simplifying assumptions, does not completely invalidate the experiment, as long as the results
are significant and these errors are within some acceptable bounds.
© 2010 Indranil Goswami. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material may be
reproduced for educational purposes.
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