Translational Dynamics

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.

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.

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.

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\(\frac{\mathrm{m}}{\mathrm{s}^2}\).

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.

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.

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.

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.

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.

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.