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Lesson Explainer: Models in Mechanics Mathematics

In this explainer, we will learn how to use a mathematical model of a real-life situation to predict and calculate distances, heights, velocities, accelerations, and forces.

In mechanical systems, the motion of objects is governed by physical principles. For instance, Earth’s gravity makes objects fall toward the ground and friction acts to slow down moving objects. Through careful observations, we can notice patterns of mechanical motions around us. When we toss a stone toward a lake, it follows a parabolic path until it lands on the surface of the water. Then, ripples of concentric circles form on the water’s surface, beginning at the point where the stone had made a splash and gradually growing outward. If we toss another stone with greater force, the same pattern ensues yet again with a greater parabola. We can expect the same motion from the stone and the lake no matter how many times we repeat the process because they must always obey the same physical laws.

Motions in the real world can be modeled by mathematical equations. Since mechanical motions repeat the same patterns with great regularity, a mathematical model allows us to understand the motion without actually conducting the experiment. This ability to simulate motion using mathematics is invaluable especially for engineers so that they can see when their designs would hold up in the real world without actually building them. Astronomers can also use these equations to understand the orbits of the Moon and the planets, which they use to predict the precise dates and times of celestial events, such as a solar eclipse. The usefulness of mathematical models is practically limitless.

In order to use a mathematical equation to model motion, we first need to understand what each of the variables of the equation represents in the real world. For instance, consider the motion of the stone we previously mentioned. We can use variables 𝑥 and 𝑦 to represent the horizontal and vertical distances, respectively, of the stone. If we are given an equation involving 𝑥 and 𝑦, its graph on the Cartesian plane would represent the trajectory of the stone. Alternatively, if we want to trace the height of the stone at any given time, we can use variables and 𝑡 to represent the height and time respectively. This would lead to a different mathematical equation compared to the first one involving the variables 𝑥 and 𝑦. Since there are different ways to model the same motion, it is important that we take note of the meaning of each variable in a given equation.

Let us consider an example where we are given a mathematical model describing an object’s motion.

Example 1: Using the Equation of a Model to Find the Displacement of a Projectile After a Time

A baseball struck by a professional baseball player travels in the air for 10 s before it lands. The motion is modeled by the equation 𝑑=20𝑡, where 𝑑 is the horizontal distance, in metres, of the baseball from the home plate 𝑡 seconds after it is hit. How far does this ball travel before hitting the ground?

Answer

In this example, we are given a mathematical equation that models the distance that a baseball travels from the home plate in terms of the time elapsed since the hit. The variables of this equation are 𝑑 and 𝑡. Let us first understand what these variables represent:

  • 𝑡 represents the time elapsed since the hit, measured in seconds;
  • 𝑑 represents the distance of the baseball from the home plate, measured in metres.

We are looking for the distance the ball travels, which is a value of 𝑑, when it hits the ground. Note that the time the ball hits the ground is given to be 10 seconds. Hence, we can substitute 𝑡=10 into the given equation to obtain the desired value of 𝑑: 𝑑=20×10=200.

Hence, the distance the baseball travels before hitting the ground is 200 m.

Le us consider another example where we are given a mathematical model for an object’s motion.

Example 2: Using the Equation of a Model to Find the Vertical Displacement of a Particle given its Horizontal Displacement

The equation =2𝑥0.2𝑥 models the motion of a soccer ball that is kicked into the air, where is the height of the soccer ball above the ground, in metres, when it travels 𝑥 metres horizontally. What is the height of the soccer ball when it travels 6 m horizontally?

Answer

In this example, we are given a mathematical equation that models the motion of a soccer ball. The variables of this equation are and 𝑥. Let us first understand what these variables represent:

  • 𝑥 represents the horizontal distance of the soccer ball, measured in metres;
  • represents the height (that is, the vertical distance) of the soccer ball, measured in metres.

Using the given mathematical model, we want to find the height of the soccer ball where the horizontal distance is equal to 6 metres. We can find this height by substituting 𝑥=6 into the given equation: =2×60.2×6=127.2=4.8.

Hence, the height of the soccer ball is 4.8 m.

In the previous example, the given mathematical model used two variables representing the horizontal and vertical distances. We can visualize this model by considering the graph of the given equation.

In the graph above, the horizontal and vertical axes represent the respective distance of the soccer ball relative to the point where the ball was kicked. In other words, this graph represents the trajectory of the soccer ball. Hence, the height of the ball when the horizontal distance is equal to 6 m is given by the vertical coordinate of the point on the graph corresponding to 𝑥=6, which equals 4.8 m, as found in the example.

When we further examine this graph, we note that the graph does not match our expectations when the horizontal distance exceeds 10 metres or when it is negative. More precisely, when 𝑥>10 or 𝑥<0, the corresponding height values are negative, indicating that the ball is below the ground. Since this is generally not possible, the mathematical model does not match our expectations of the real world. Hence, the height predicted by this model is only likely to be valid when 0𝑡10.

In mathematical modeling, we often encounter such situations where the model (that is, the equation) only makes sense within a certain range of values. This range, which is sometimes referred to as the physical domain of the model, is an essential part of the mathematical model and is generally indicated when the equation is provided. In some cases, we can deduce this range from context by excluding all unreasonable values given by the equation.

In the next example, we will identify the range of values for which the given mathematical model is valid.

Example 3: Finding the Time Range for which a Model of a Projectile Is Valid

The motion of a stone from a rooftop is modeled by the equation =3𝑡+12𝑡+36, where is the height of the stone above the ground, in metres, 𝑡 seconds after the stone is tossed. Find the largest range of values for 𝑡 for which this model can be valid.

Answer

Recall that mathematical models of mechanical motions are often only valid within a certain range of values for which the models are intended. When the model produces values that are unreasonable for the given context, we can know that the model ceases to be valid. Hence, we can find the largest range of values of variable 𝑡 for which the model is valid by first identifying the range of 𝑡 for which the model produces unreasonable values.

We are given an equation that models the motion of a stone. The variables of this equation are and 𝑡. Let us first understand what these variables represent:

  • 𝑡 represents time elapsed since the toss, measured in seconds;
  • represents the height of the stone above the ground, measured in metres.

We can visualize this model by looking at the following graph.

Firstly, we know that this motion began when the stone was tossed. Since 𝑡 represents time after the toss, this model’s values are not reasonable for our given context when 𝑡<0 (that is, before the toss).

Secondly, we can consider the values for variable produced by this model. Since represents the stone’s height above the ground, it cannot take negative values. In the graph, we can see that values of become negative after some point, which is unrealistic. The value of 𝑡 where this occurs is a root of this equation, which can be found by setting =0. This leads to 0=3𝑡+12𝑡+36.

We can find the roots of this equation by factoring. Dividing both sides of the equation by 3, we get 𝑡4𝑡12=0.

We can factor this equation: (𝑡6)(𝑡+2)=0.

This leads to the roots 𝑡=6 and 𝑡=2. The negative root, 𝑡=2, is the root on the left side of the vertical axis, which is already not relevant since it occurred before the toss. Hence, the value of 𝑡 after which the model is invalid is equal to 6.

We have found that the model produces unrealistic values when 𝑡<0 or 𝑡>6. This means that the largest range of values for 𝑡 for which the model can be valid is 0𝑡6, or [0,6].

Mechanical motions are affected by many real-life factors, many of which are usually negligible. For instance, let us consider what real-life factors may affect the motion of the stone given in the previous example. An important factor at play here is Earth’s gravity, which pulls the stone downward. As a consequence, the graph of the equation from this example demonstrates that the stone travels down toward the ground after peaking at the top of the parabola. However, Earth’s gravity varies depending on the distance of an object from the center of Earth. This means that the magnitude of gravity acting on the stone would be different every time its height, the distance from Earth’s center, changes. While this is technically true, this variation in gravity is minuscule because the change in the stone’s height is very small relative to its distance from Earth’s center. Hence, assuming the gravity to be constant over the course of this motion is not likely to lead to significant error.

Definition: Modeling Assumptions of Gravity

In most mechanical systems, gravity is assumed to be a constant downward force whose magnitude is proportional to the object’s mass. More specifically, the acceleration of an object due to gravity is denoted as 𝑔, where 𝑔=9.8/ms.

Assuming Earth’s gravity to be constant is an example of modeling assumptions. Making assumptions is an important step in creating mathematical models of real-life situations. Ignoring the effects of negligible real-life factors can lead to simpler models still containing the core aspects of the motion. Such simplified models are said to be ideal, or idealized, since we are imagining the motion to occur within an ideal world where these minuscule factors do not exist. While ideal models do not exactly represent the real-life situation, their results can still be meaningful.

Modeling assumptions are often indicated by specific terminologies used when describing the mathematical model. Let us consider some frequently used terminologies related to dimensions of objects.

Definition: Particles, Rods, and Lamina

A particle is an object whose mass is concentrated at a single point. In other words, a particle has no dimensions (length, width, or thickness); hence, it has zero volume. When assuming that an object is a particle, we ignore rotational forces and air resistance in the object’s motion.

A rod is an object whose mass is concentrated on a line, which cannot be bent or cut. A rod has a nonzero length, but its other dimensions are equal to zero. When assuming that an object is a rod, we ignore air resistance in the object’s motion.

A lamina is an object whose mass is concentrated on a flat surface, which cannot be deformed. A lamina has a nonzero surface area, but its thickness, hence its volume, is equal to zero.

Particles, rods, and laminae cannot exist in the real world since the volumes of these objects are equal to zero. In the real world, something with zero volume must also have no mass. This is because mass is a measure of how much matter there is in an object, and all matter takes up a nonzero amount of volume. However, we can imagine an ideal world where these objects do exist and are still subject to the relevant physical laws. Hence, mathematical models can be constructed based on these assumptions.

Such assumptions are beneficial when the object’s volume is likely to have negligible effects on the motion. For instance, the volume of the stone tossed from a rooftop, as given in the previous example, is not likely to play a major part in its height as it falls to the ground. Because a typical stone is a round and dense object, air resistance would also be negligible in comparison to the gravity, which plays a larger part. In this case, it would be reasonable to assume that the stone is a particle in this mathematical model.

In the next example, we will determine in which of the given situations the modeling assumption of a particle is not appropriate.

Example 4: Understanding the Effects of Modeling Assumptions

In which of the following contexts is it not appropriate to assume that the object is a particle?

  1. A soccer ball rolling down a hill
  2. An open parachute falling through the air
  3. A baseball traveling in the air
  4. A satellite moving in an orbit
  5. An ice skater moving on ice

Answer

Recall that, in mathematical models of mechanical motion, a particle is an object whose mass is concentrated at a single point. A particle has no length, width, or thickness, but it has a nonzero mass. Such an object is impossible to exist in the real world. When we say that an object is a particle, we are making an idealistic assumption to simplify the model where the dimensions (or volume) of the object have negligible effects on its motion. In particular, under this modeling assumption, we ignore any rotational forces and air resistance in the object’s motion, which result from its dimensions. However, this assumption would not be valid when the object’s dimensions have a significant effect on its motion.

Let us consider each of the given contexts to determine whether or not such assumption is valid.

  1. When modeling a soccer ball rolling down a hill, we notice that its rotations and surface friction can affect its motion. However, such effects are minuscule relative to the one associated with Earth’s gravity, which is generating the downward motion of the soccer ball. We can simplify the model to be an object of the same mass, which is traveling down the hill without any rotational or frictional effect. While this simplified model would not be an exact representation of the soccer ball, it still captures the most important aspects of the motion. Hence, it is appropriate in this context to assume that the soccer ball is a particle.
  2. An open parachute falling through the air experiences great air resistance due to its large surface area. Without the surface area of the parachute, and hence without the air resistance, the open parachute would fall rapidly to the ground, accelerated by Earth’s gravity. Such assumption is not appropriate in this context because it significantly changes the nature of the object’s motion.
  3. A baseball traveling in the air is affected by many real-life factors, such as its rotations and air resistance. However, these effects are small relative to Earth’s gravity and the initial force, which initiated the baseball’s motion. We can simplify the model by considering an object of the same mass traveling through the air without the effects of rotation and air resistance. While this simplified model would not be an exact representation of the baseball, it still captures the most important aspects of the motion. Hence, it is appropriate in this context to assume that the baseball is a particle.
  4. When a satellite travels in an orbit in outer space, it experiences little to no air resistance. Instead, its motion is mainly controlled by Earth’s gravity. The assumption that the satellite has no dimensions (or volume) would not significantly alter its motion. Here, we should note that “rotating in orbit” refers to the object traveling in a circular (or elliptical) trajectory around Earth, rather than the satellite spinning in place (which would be referred to as “revolving”), which would be neglected when assuming that it is a particle. Hence, it is appropriate in this context to assume that the satellite is a particle.
  5. An ice skater moves on ice with negligible effects from surface friction or air resistance. We can simplify the model by comparing it to the motion of an object of the same mass traveling on ice without the effects of rotation and air resistance. Hence, it is appropriate in this context to assume that the ice skater is a particle.

This leads to option B, an open parachute falling through air, which should not be modeled under the assumption that it is a particle.

We have discussed modeling assumptions pertaining to the dimensions of an object. There are many other modeling assumptions and terminologies that we should be familiar with. In particular, mathematical models often contain simplifying assumptions for the components of mechanical systems, such as strings, wires, pulleys, pegs, and floors. Strings and wires are used to connect two or more objects, pulleys are used to change the direction of force, and floors are used as boundaries for motions in mechanical systems. In terms of dimensionality, pulleys often are assumed to be dimensionless like particles, strings and wires only have lengths like rods, and floors only have surface areas like laminae. Unlike particles, rods, and laminae, these objects’ roles are limited to assisting in the motion of the objects. To prevent these objects from interfering with other aspects of motion, a number of modeling assumptions are used. A few of these assumptions are represented by the adjectives given below.

Definition: Adjectives of Modeling Assumptions for Components of Mechanical Systems

A rigid object is one that cannot be deformed (that is, bent, cut, or punctured). Wires and floors are generally assumed to be rigid, while strings are generally not rigid.

A light object is one with zero mass. Strings, wires, and pulleys are generally assumed to be light.

An inextensible object is one that cannot be stretched or contracted. Lengths of inextensible wires or strings remain constant throughout the motion, regardless of the load they bear.

A uniform object is one whose density is constant throughout its body. Strings and wires are generally assumed to be uniform.

A fixed object is one that remains in one place and cannot be moved. Strings or wires may be fixed on one or both ends. Pegs are assumed to be fixed.

A smooth object is one that offers no friction. An object that is not smooth is said to be rough.

As with the modeling assumptions introduced earlier, these modeling assumptions only exist in ideal worlds. A pulley cannot be truly weightless nor can a string bear heavy loads without any extension. However, if the weight of a pulley used in a mechanical system is small in comparison to other objects at play, we can assume the pulley to be light. Similarly, a string used in a typical pendulum can be assumed to be inextensible since the string’s length does not vary significantly during the pendulum’s motion. Also, while all floors provide friction to moving objects that come in contact with them, we can assume that a floor is smooth if the friction does not significantly affect the motion.

In our final example, we will consider which of the modeling assumptions above are valid in a given context.

Example 5: Understanding the Effects of Modeling Assumption

Which of the following assumptions is not appropriate when constructing a mathematical model representing a mechanical system where a pulley system drags a heavy box on a concrete floor?

  1. The pulley is smooth.
  2. The pulley is light.
  3. The pulley is fixed.
  4. The string is inextensible.
  5. The floor is smooth.

Answer

Recall that, when modeling real-life situations, we notice that modeling assumptions are made to simplify the context. These are idealistic assumptions that have little impact on the motion under interest. In this example, we are given a mechanical system where a pulley is used to pull a heavy box on a concrete floor. Each option associates an adjective with a component of this system. These adjectives represent specific modeling assumptions on the associated component. Let us recall the implications of these assumptions given in the options and determine whether or not they are appropriate for the given context.

  1. A smooth object does not produce friction for objects in motion that come in contact with it. In this mechanical system, the string comes in contact with the pulley. As the string moves, the pulley does cause a little friction, but the magnitude of this friction is much smaller in comparison with the friction between the heavy box and the floor. Since this friction is negligible, we can assume idealistically that the pulley does not cause any friction in order to obtain a simpler model. Hence, this assumption is appropriate for the given context.
  2. A light object is one that has no mass. While it is impossible for the pulley, or any matter in the real world, to be massless, the mass of the pulley is likely to be much smaller in comparison to that of the heavy box. Also, the mass of the pulley does not play any part in this motion. Since we can ignore the mass of the pulley without affecting the motion, this assumption is appropriate for the given context.
  3. An object fixed in place cannot be moved. We can typically assume that a pulley in a system will be stationary since its purpose is to support a string or cable. Hence, this is a reasonable assumption for the given context.
  4. An inextensible object is one that cannot be stretched or contracted. In our model, the string is pulling the heavy box. Any string in the real world would stretch when heavy loads are applied while the amount of this stretch would vary depending on the material. A typical pulley system would use an inflexible string, so the string’s stretching would be negligible in comparison to the original length of the string. To simplify the model, we can assume that the string does not stretch at all, which would not affect the motion significantly. Hence, this is a reasonable assumption for the given context.
  5. Unlike the pulley, there will be significant friction between the floor and the heavy box due to the box’s weight and the material of the floor (concrete). This friction significantly affects the motion since the box would move effortlessly along the floor otherwise. Ignoring the friction would alter the nature of this motion. Hence, it is not appropriate in the given context to assume that the floor is smooth. Instead, we should assume that the floor is a rough surface, which is the opposite of a smooth surface.

This leads to option E, which states that the floor is smooth.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • To use a mathematical equation to model motion, we first need to understand what each of the variables of the equation represents in the real world.
  • In general, mathematical models are only valid within a certain range of values, known as the physical domain of the model. In some cases, we can deduce this range from the context by excluding all unreasonable values given by the equation.
  • Modeling assumptions are important in constructing simplified mathematics models. While these are idealistic assumptions that are not possible in the real world, the resulting models are often still meaningful. The following terminologies are used to represent modeling assumptions.
    • In most mechanical systems, gravity is assumed to be a constant downward force whose magnitude is proportional to the object’s mass. More specifically, the acceleration of an object due to gravity is denoted as 𝑔, where 𝑔=9.8/ms.
    • A particle is an object whose mass is concentrated at a single point. When assuming that an object is a particle, we ignore rotational forces and air resistance in the object’s motion.
    • A rod is an object whose mass is concentrated on a line, which cannot be bent or cut. When assuming that an object is a rod, we ignore air resistance in the object’s motion.
    • A lamina is an object whose mass is concentrated on a flat surface, which cannot be deformed. A lamina has a nonzero surface area, but its thickness, hence its volume, is equal to zero.
    • A rigid object is one that cannot be deformed (that is, bent, cut, or punctured). Wires and floors are generally assumed to be rigid, while strings are generally not rigid.
    • A light object is one with zero mass. Strings, wires, and pulleys are generally assumed to be light.
    • An inextensible object is one that cannot be stretched or contracted. Lengths of inextensible wires or strings remain constant throughout the motion, regardless of the load they bear.
    • A uniform object is one whose density is constant throughout its body. Strings and wires are generally assumed to be uniform.
    • A fixed object remains in one place and cannot be moved. Strings or wires may be fixed on one or both ends, and pegs are assumed to be fixed.
    • A smooth object is one that offers no friction. An object that is not smooth is said to be rough.

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