Resistive Force

Types of Resistive Forces

Sliding friction is an example of a resistive force and is caused when two bodies attempt to slide over each other. There is an opposition to the direction of motion of the bodies due to this friction. Air resistance (or drag) is another example of a resistive force where an object is slowed down when moving through the air, which is the medium. Air resistance is classified as a type of fluid friction. The resistive force is always opposite to the direction of motion of the object relative to the medium.

The gravitational force can also be considered a resistive force if it is opposing the vertical takeoff of a rocket, for example. Furthermore, the normal force is also a resistive force. Imagine an object at rest on a horizontal surface. The force due to gravity (weight) acts to attract the object towards the center of the Earth. However, the normal force resists this with an equal and opposite force, causing the object to remain stationary on the surface.

Resistive Force of Friction

We had defined the resistive force as one that opposes the motion of an object due to the properties of the medium through which it travels. One of the most well-known of these resistive forces is friction. We will focus on static friction and kinetic friction.

Frictional Force

The frictional force occurs between two solid objects that slide over each other. The relative motion between the surfaces of the two objects exerts a force on each object that is parallel to both surfaces and opposing each of their motions. The frictional force depends on the material of each object and how much they are pressed together.

As an example, imagine sliding a laminated book across a relatively smooth table top. The book will slide across the table fairly easily before coming to rest; the frictional force exists but is not great. Next, if you were to add a heavy weight on top of the book and again push it across the table, it would come to rest sooner as the frictional force between its surface and the table's surface has increased.

We will discuss the types of frictional forces later on, but the general equation for the friction force \(F_\mathrm{friction}\) is,

\[F_{\mathrm{friction}}=\mu N,\]

where \(\mu\) is the coefficient of friction of the surface along which the object is moving and \(N\) is the normal contact force between the surface and the object.

Static Frictional Force

The static frictional force occurs when the object and surface are at rest relative to each other. There is no motion of one relative to the other. The object will remain at rest until a force is applied that is greater in magnitude than the maximum static frictional force \(f_s\) that is given by \[f_{s}^{max}=\mu_{s} N,\] where \(\mu_s\) is known as the coefficient of static friction of the surface and \(N\) is the normal contact force between the object and surface.

Kinetic Frictional Force

The kinetic frictional force \(f\) for a moving object of mass \(m\) is given by the following equation, \[f_k=\mu_{k}N,\] where \(\mu_{k}\) is known as the coefficient of kinetic friction and \(N\) is the magnitude of the normal reaction force that the surface applies on the object. Note that the kinetic frictional force could be a rolling or sliding frictional force.

Viscous Force

Viscous forces occur when solid objects move through fluids. The solid object moves layers of the fluid as it passes through it. The relative motion between two layers of the fluid provides a resistive force to the motion of the object. The viscous force depends on a property of the fluid called viscosity. The greater the viscosity of the fluid, the greater the viscous force that will be applied to an object moving through it. The magnitude of the viscous force \(F_{\mathrm{viscous}}\) is given by the equation,

\[F_\mathrm{viscous}=\alpha v,\]

where \(\alpha\) is a constant that depends on the viscosity and other physical properties of the fluid and \(v\) is the speed of the object moving through the fluid.

Drag force

A drag force occurs when an object moving through a fluid needs to push particles of the fluid out of its way. It exerts a force on the particles comprising the fluid and hence feels a force back from those particles. The force on the object opposes its motion and is hence resistive. The drag force is proportional to the square of the object's velocity \(v\) and its magnitude is given by \[F_{\mathrm{drag}}=\beta v^2,\] where \(\beta\) is the drag coefficient of the fluid.

Resistive Force Formula

We are going to consider the case of an object moving at a low speed through a medium. We will think of a scenario in which an object moves with velocity \(\vec{v}\) relative to some medium in which it feels a resistive force \(\vec{F}_r.\) In this case, the resistive force is directly proportional to the velocity of the object and in the opposite direction, that is,

\[\vec{F}_r \propto \vec{v}.\]

We need to include a constant in this expression to turn this into an equation. The equation for the resistive force, in terms of the velocity, becomes \[\boxed{\vec{F}_r=-k\vec{v},}\] where \(k\) is the aforementioned constant which depends on the physical properties of the object and the medium it travels through. The negative sign here indicates that the resistive force \(\vec{F}_r\) is in the opposite direction to the velocity \(\vec{v}.\) This resistive force is clearly greater as the object's velocity increases through the medium. In the case of cycling against the wind, we find this task significantly more difficult as our speed increases.

Note that for heavy objects or objects moving at a high speed, the resistive force is directly proportional to the square of the speed, \[\vec{F_r} \propto v^2,\] this would be the case for a skydiver.

Terminal velocity

Let us consider an example of a slow-moving object; a feather falling from a tree to the ground slowly. The feather will at some point reach a constant speed whilst falling toward Earth. If the resistive force is directly proportional to the velocity of the feather, it is zero when the feather's speed is zero; at the start of its descent. At this point, the only force acting on the feather is its weight \(m\vec{g}\) and the magnitude of its acceleration is \(g.\) As its speed increases from zero, it starts to experience a resistive force \(\vec{F}_r\) in the opposite direction due to air resistance as it moves through this medium. The net force on the feather can be given as \[\begin{align}\vec{F}_{\text{net}}&=m\vec{g}+\vec{F}_r,\\|\vec{F}_{\text{net}} |&=mg-kv,\end{align}\]

which can be used to find the net acceleration \(a\) on the feather at any time using Newton's second law,

\[\begin{align}ma&=mg-kv,\\a&=g-\frac{k}{m}v.\end{align}\]

Remember that the resistive force is increasing with speed, consequently, the net force is constantly decreasing and, from the equation above, so too is the net acceleration. When the feather reaches a point at which its speed is constant, we say it has reached terminal velocity \(v_T\). The feather no longer accelerates and we can find the terminal velocity from the equation above by setting \(a=0\), \[\begin{align}0&=g-\frac{k}{m}v_T,\\v_T&=\frac{mg}{k}.\end{align}\] We can visualize this scenario with the help of the image below.

A graph of velocity vs time for an object experiencing air resistance whilst falling can be seen in Fig. 2. below.

Fig. 2 - A graph of velocity against time for a falling object experiencing air resistance, showcasing the terminal velocity

The rate of change of velocity (acceleration) is greatest at the start of the fall and decreases gradually as the resistive force increases in magnitude and opposes the weight. The object reaches terminal velocity when the weight is equal in magnitude to the resistive force.

Resistive Force Examples

Let's go back to the example of the falling feather in the section on terminal velocity. We can try to write the formula for the acceleration in a way that makes the velocity the subject of the formula,

\[\begin{align}a&=g-\frac{k}{m}v,\\\frac{\mathrm{d}v}{\mathrm{d}t}&=g-\frac{k}{m}v. \end{align}\]

This is a separable first-order differential equation that can be solved using multiple methods, one of which requires substitution. Doing so gives us the following, \[\begin{align}\frac{\mathrm{d}v}{g-\frac{kv}{m}}&=\mathrm{d}t, \end{align}\] from which we can let, \[\begin{align}u&=g-\frac{kv}{m},\\ \Rightarrow \mathrm{d}u&=-\frac{k\,\mathrm{d}v}{m},\\ \Rightarrow \mathrm{d}v&=-\frac{m\,\mathrm{d}u}{k}.\end{align}\] Our new differential equation becomes \[-\frac{m}{k}\int_g^{g-bv/m} \frac{\mathrm{d}u}{u}=\int_0^t \mathrm{d}t',\] which gives us the following solution for the velocity \(v\) as a function of time \(t\), \[v(t)=\frac{mg}{k}\left(1-e^{-kt/m}\right).\] Here we assume that the object's speed \(v=0\) when the time \(t=0\), which is true for the feather falling from rest. Note that for a fluid, the viscosity of fluid provides the resistive force on the object. Let's test our new-found knowledge by considering the following example.

Let's consider another example to see if the object in the previous example has reached terminal velocity.

Work Done Against Resistive Forces

Recall that when forces are involved there are usually energy transformations occurring, which in turn means that work will be done on an object. The work done on an object \(W\) by an applied force \(F\) is the sum of the gain in kinetic energy \( \Delta E_K \) of the object and the work done by the object against the resistive force \(W_r\). That is,

\[\begin{align}W&=\Delta E_K+W_r,\\W_r&=W-\Delta E_K.\end{align}\] The work done on the object by the applied force increases its kinetic energy. The work done by the object against the resistive force takes energy away from the object, decreasing its kinetic energy.