Why do things move? It seems such a profound question for something that happens around us all the time. In fact, it's something that puzzled scientists and philosophers for hundreds of years. For example, Aristotle believed that all objects had a 'natural' place to which objects fell towards depending on which 'element' they were made out of. However, it wasn't until Newton's laws of motion that an understanding of forces as the causes of motion was systematically laid out. In this article, we're going to look at translational dynamics which analyses forces and how they produce a type of motion known as translational motion. Before looking at translational dynamics, let's recap Newton's law of motion.
Newton's Laws of Motion
Newton's laws of motion are three fundamental laws that describe the forces acting on an object and its motion. Sir Isaac Newton published his principles of motion in Philosophiae Naturalis Principia Mathematica on July 5, 1687. These simple laws served as the foundation of classical mechanics, and Newton himself utilized them to describe a wide range of phenomena relating to the motion of physical objects.
Newton's First Law
Newton's first law is concerned with the idea of inertia. It states that an object will remain at rest or in uniform (i.e at constant velocity) motion in a straight line unless it is acted upon by a net external force. It states that objects don't just start moving, or change velocity, of their own accord, there must be a force involved. Alternatively, we can summarise the law as 'no acceleration means there is no force.
Newton's First Law - An object will remain at rest or in uniform motion in a straight line (i.e at constant velocity) unless it is acted upon by a net external force.
It's important to remember that because velocity is a vector with a direction, changing direction is a form of acceleration. This is why the law states that an object 'remains in a straight line unless acted upon by an external force. This idea prompted Newton to suggest that the planets orbiting the sun experience the force of gravity.
Fig. 1 - A box on the ground will stay stationary until an external force is applied. This is because the weight of the box and the normal force cancel each other, meaning the net force is zero.
If an object is stationary this doesn't necessarily mean that no forces are acting on it, simply that the sum of all the forces, the net force, is zero. A stationary box is affected by both its weight and the normal force of the floor pushing up on it. Because their vector sum equals zero, these two forces cancel each other out, and the net force is zero. That's why the box doesn't just fall through the floor!
Newton’s Second Law
Of all the three laws of motion, Newton's second law is the one that gives a direct mathematical relationship between the motion of an object and the force it experiences. It will probably be the single law of physics that you use most in all your studies.
Newton's Second Law - The resultant force acting on an object is equal to the product of the object’s mass and its acceleration.
This is expressed mathematically by the famous equation:
\[F=ma\]
where \(F\) is the resultant or net force measured in Newtons \(\mathrm{N}\), \(m\) is the mass of the object in kilograms \(\mathrm{kg}\), and \(a\) is the acceleration of the object in meters per second squared\(\frac{\mathrm{m}}{\mathrm{s}^2}\).
If a box with a mass of 5 kg is pulled with a force of 25 N, what is the subsequent acceleration of the box?
Solution
According to Newton’s second law:
\[\text{Resultant Force}=\text{mass}\cdot\text{acceleration}\]Here, the box is pulled with a force of 25 N, which is our resultant force, and the mass is given to us in the question. If we plug these into the equation for Newton’s second law, we can find the magnitude of the acceleration of the box.\[\begin{align}25\,\mathrm{N}&=5\,\mathrm{kg}\cdot a\\\Rightarrow a&=\frac{25\, \mathrm{N}}{5\,\mathrm{kg}}\\&=5\frac{\mathrm{m}}{\mathrm{s}^2}\end{align}\]
Newton's first law is a special case of the second law, as clearly if the acceleration of an object is zero then there is resultant force is also zero.
Whilst the definition of Newton's second law given above is more common and is almost always sufficient for solving problems there is a more rigorous and fundamental definition that is useful to know. It states that the resultant force on an object is equal to the time derivative of the object's momentum. Mathematically,\[F=\frac{\mathrm{d}p}{\mathrm{d}t}\]
Recall that the momentum of an object is \(p=mv\). This formulation can be used in situations where the mass of the object is non-constant and is more precise as mass is not necessarily a conserved quantity whereas momentum always is.If the mass of an object is a constant then we easily recover the usual equation for Newton's second law\[\begin{align}F=&\frac{\mathrm{d}p}{\mathrm{d}t}\\=&\frac{\mathrm{d}(mv)}{\mathrm{d}t}\\=&m\frac{\mathrm{d}v}{\mathrm{d}t}\\=&ma\end{align}\]The final step comes from the definition of acceleration as the time derivative of velocity.
Newton's Third Law
The third and final Newton's law of motion concerns reaction forces. Imagine that you are sitting in a wheelchair, and you push against a wall causing you to roll backward. Why is this? Well when you push against the wall, the wall exerts an equal and opposite reaction force on you causing you to roll backward. This idea is encapsulated by Newton's third law. We call this combination of action and reaction forces, action-reaction pairs.
Newton's Third Law - For every force, there is always a reaction force with the same magnitude acting in the opposite direction.
It is worth noting that the action-reaction pairs do not apply to the same object and are of the same type of force. For example, a book resting on a table experiences a gravitational force from the earth, the reaction force is the gravitational force exerted on the earth by the book.
When a ball bounces on the ground it exerts a downwards force onto the ground, the ground then exerts a force of equal magnitude onto the ball upwards in the opposite direction. This reaction force causes the ball to bounce upwards. This may seem strange as we never see the earth move downwards when we bounce a ball, however, remember that \(F=ma\), so the huge mass of the earth means that the acceleration of the earth is negligible.
Translational Dynamics Meaning
When a body's position does not change with respect to time, we say it is at rest. However, when a body's position changes with respect to time, we say it is in motion. The study of motion in physics falls into two categories dynamics and kinematics. In kinematics, we are only concerned with things like an object's position, velocity, acceleration, etc. and how it changes over time. Kinematics does not look at the causes of motion and so ignores quantities such as momentum, force, or energy. On the other hand, dynamics looks at the broader picture looking at why an object moves and analyzing motion by looking at the resultant forces on an object or the work done by the object. Newton's Laws of Motion give us an understanding of forces as the cause of motion, from these laws we can then build up a full understanding of the motion of an object. We define two kinds of dynamics, translational dynamics, and rotational dynamics. Rotational dynamics is concerned with objects moving around an axis of rotation such as spinning objects or objects undergoing circular motion.
Fig. 4 - The motion of a Ferris wheel is rotational and falls under rotational dynamics, Wikimedia Commons
However, in this article, we will concentrate on translational dynamics.
Translational dynamics concerns the motion of objects where all parts of the body travel uniformly in the same direction. We can think of it as a kind of sliding where the orientation of the boy does not change whilst the object moves.
For example, a bullet fired from a gun undergoes translational motion as does a block sliding down an inclined plane.
Translational Dynamics Model
Being able to model the motion of an object or group of objects is the central purpose of translational dynamics. In physics, we call an object, or group of objects, under consideration a system. A system can be really simple like a single non-interacting particle, or it can be as complex as a galaxy held together by gravitational interactions.
In translational dynamics, we can simplify things by considering a system's mass to be accumulated entirely within the center of mass, this way forces only act at one point of the system and we don't have to consider how the force acts on each component of the system. We can make this simplification when the properties of the constituent particles are not important to model the behavior of the macroscopic system.
The center of mass of a system is the point of a system where the weighted relative position of the distributed mass of the system is zero. For a system composed of a finite number of point masses, it can be found using the following formula\[\vec{r}_{\mathrm{cm}}=\frac{\sum{m_i\vec{r}_i}}{\sum{m_i}}\]If the distribution of mass is continuous, like in a single solid object, then we need to integrate instead
\[\vec{r}_{cm}=\frac{\int\vec{r}\mathrm{d}m}{\int \mathrm{d}m}\]
If a force acts through an object's center of mass it will experience linear translational motion, it's only if the force acts away from the center of mass that rotational motion can occur. This is why objects balance if they're held up at their center of mass.
Fig. 5 - By holding up the toy at its center of mass the toy can balance on the person's finger
In translational dynamics, one of the most commonly occurring problems is calculating the resultant force on an object to understand the direction of an object's acceleration. This involves adding up all the force vectors acting on the object and finding the resultant force vector. Keeping track of all the forces acting on a body is easiest done using a free-body force diagram. These are simple diagrams where each force acting on an object is represented by an arrow pointing in the direction the force acts and accompanied by the magnitude of the force written beside it. Such a diagram is shown below, demonstrating the forces acting on a block sliding down an inclined plane.
Fig.6-Free-body diagram of a block on an incline plane. The friction force acts opposite to the direction of the object's motion, whilst the normal force acts perpendicularly to the surface and the weight acts downwards from the centre of mass.
When multiple forces are acting on an object that are not acting along more than one axis, then we need to select a coordinate system to resolve the forces into their components. By adding the components we can find the resultant force in this coordinate system. In the diagram above, the three forces on the block are not acting in the same direction, so we need to resolve the vectors into their \(x\) and \(y\) components to find the resultant force vector.
Notice that the angle between the weight and the normal force is equal to the angle of inclination \(\theta\). Choosing to resolve the forces into components defined by a coordinate system whose axes are parallel and perpendicular to the surface of the ramp means that we only need to resolve the weight into its components. The friction and normal force are already aligned with the axes of this coordinate system simplifying the calculation. Resolving the weight into these components gives:\[\begin{align}W_x=& mg\sin(\theta)\\W_y=& mg\cos(\theta)\end{align}\]
This model of forces and resolved force components set us up to calculate the resultant force acting on the object and its acceleration.
Translation Dynamics Formulas
Once all the forces have been drawn on a free-body diagram and an appropriate coordinate system has been chosen to resolve the force vectors into, we can use Newton's laws to calculate the acceleration of an object.The formula for calculating the resultant force \(F_{\text{net}}\) can be given as
\[\begin{align}(F_{\text{net}})_x&=\sum_i (F_i)_x\\(F_{\text{net}})_y&=\sum_i (F_i)_y\end{align}\]where \(i\) is an index for the forces acting on the object.
Returning to the block on the inclined plane, labeling the friction as \(F_{\mu}\) and the normal force \(F_{\text{norm}}\) the resultant force is given by
\[\begin{align}(F_{\text{net}})_x&=F_{\mu}+mg\sin(\theta)\\(F_{\text{net}})_y&=F_{\text{norm}}+mg\cos(\theta)\end{align}\]
We can use Newton's third law of motion to find the value of the normal force acting on the block. As every action has an equal and opposite reaction force, the normal force must be the reaction force of the plane's surface equal and opposite to the component of the weight acting on the surface.
\[\begin{align}&F_{\text{norm}}=-mg\cos(\theta)\\&\Rightarrow (F_{\text{net}})_y=F_{\text{norm}}+mg\cos(\theta)=0\end{align}\]
This is as we expect as we chose our \(y\)-axis to be perpendicular to the surface of the inclined plane, and clearly, the block is neither falling through the plane nor levitating above it! If we had instead lined up our \(y\) axis to be perpendicular to the ground, there would be a force acting downwards.The block is sliding down the plane, so we need to use Newton's second law to find the acceleration of the block along the \(x\) axis. Recall that\[F_{\text{net}}=ma\]so, the acceleration along the \(x\) axis is\[\begin{align}F_{\mu}+mg\sin(\theta)&=ma_x\\\Rightarrow a_x&=\frac{F_{\mu}}{m}+g\sin(\theta)\end{align}\]
The value of \(F_{\mu}\) is determined by the surface's coefficient of friction \(\mu\)and is given by
\[F_{\mu}=\mu F_{\text{norm}}=\mu m\cos(\theta)\]
This gives a full run-through of how we can use Newton's laws of motion to determine the translational dynamics of an object, so let's apply this method to an explicit example.
Consider a wooden block held up by two cables, as shown in the diagram below, with the wooden block having a mass of \(5.00\mathrm{kg}\). If the first cable is at an angle of \(\theta_1=-25.0^{\circ}\) to the vertical, and the second is at an angle of \(\theta_2=45.0^{\circ}\) what will the tension in each cable be?
Fig.7- By resolving the forces acting on the block we can find the tension in each cable required to hold it up.
First write down all the forces acting on the block. Let the tension in cable one be \(F_1\) and in cable two \(F_2\) and obviously the weight will be \(F_{weight}=mg=5.00\cdot9.81=49.0\,\mathrm{N}\). By choosing our coordinate axes such that the \(y\) axis is the vertical axis and \(x\) is the horizontal, we know that, if the block is in equilibrium, the following equations must hold.
\[\begin{align}(F_1)_x+(F_2)_x&=0\\(F_1)_y+(F_2)_y&=F_{\mathrm{weight}}=49.0\,\mathrm{N}\end{align}\]
We can express these components using trigonometry\[\begin{align}F_1\sin(\theta_1)+F_2\sin(\theta_2)&=0\,\mathrm{N}\\F_1\cos(\theta_1)+F_2\cos(\theta_2)&=49.0\,\mathrm{N}\end{align}\]
Re-arranging the first equation and plugging in the angles gives:
\[F_1=-F_2\frac{\sin(-25^{\circ})}{\sin(45^{\circ})}=0.60F_2\]
Substituting this into the second equation
\[\begin{align}0.60&\cdot\cos(-25.0^{\circ})F_2\,\mathrm{N}+F_2\cos(45.0^{\circ})\,\mathrm{N}=49.0\,\mathrm{N}\\\Rightarrow& 0.60\cdot(0.91)F_2\,\mathrm{N}+0.71F_2\,\mathrm{N}=49.0\,\mathrm{N}\\\Rightarrow& F_2=\frac{49}{0.60\cdot0.91+0.71}\,\mathrm{N}=39.0\,\mathrm{N}\end{align}\]This means the two tensions are
\[\begin{align}F_2&=39.0\,\mathrm{N}\\F_1&=0.60F_2=23.0\,\mathrm{N}\end{align}\]
Translational Dynamics - Key takeaways
In mechanics, an object or group of objects under consideration is called a system
Kinematics looks at how a system moves over time through quantities such as position velocity and acceleration without considering the causes of motion, whereas dynamics considers analyses the causes of motion and is concerned with things like force and energy
When a body moves as a whole and every portion of the body travels the same distance in the same amount of time, then we say the body is in translational motion. On the other hand, if the body is rotating around a fixed axis, this is called rotational motion.
Newton's Laws are the fundamental laws governing the relationship between forces and motion
Newton’s first law states that if an object is at rest or in uniform motion unless there is an external force, it will preserve its status.
Newton’s second law, force is the product of mass and acceleration.
Newton’s third law states that for every action force, there is a reaction force of the same magnitude but in the opposite direction.
We can use free body diagrams to visualize how forces act on a system and to find the resultant force acting on the system's center of mass
References
- Fig. 1 - A box on the ground, Vaia Originals
- Fig. 2 - Person pushing wall, Vaia Originals
- Fig. 3 - Bouncing ball, Vaia Originals
- Fig. 4 - Ferris Wheel (https://commons.wikimedia.org/wiki/File:Ferris_Wheel_-_2590239345.jpg), by seabamirum licensed by CC BY 2.0 (https://creativecommons.org/licenses/by/2.0/deed.en)
- Fig. 5 - Bird Toy Showing Center of Gravity (https://commons.wikimedia.org/wiki/File:Bird_toy_showing_center_of_gravity.jpg) by APN MJM licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/deed.en)
- Fig. 6 - Free body force diagram, Vaia Originals
- Fig. 7 - Cable and block, Vaia Originals
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