Particle kinetics

Classical mechanics

Newton's equations relate the acceleration $ \vec{a} $ of a point mass with mass $ m $ to the total applied force $ \vec{F} $ on the mass (sum of all applied forces). They are:

Newton's equations:

$$ \vec{F} = m\vec{a} $$

There is no derivation for Newton's equations, because they are an assumed model for dynamics. We can only verify them by comparing with experimental evidence, which confirms that Newtonian dynamics are accurate for non-relativistic and non-quantum systems.

A point mass moving in the plane with an applied force. You can try to made the mass move in a circle and then see what happens when the force is suddenly removed, which will demonstrate Newton's first law (no net force implies motion at constant speed in a constant direction). Also observe which force directions cause the speed to increase or decrease.

Click and drag to impart a force on the particle.

Application alert!

Accelerating and braking uses particle kinetics.

Particle kinetics can be used to analyze how cars accelerate and brake.

Application alert!

Banked turns uses particle kinetics.

Particle kinetics can be used to analyze how vehicles move along banks.

Application alert!

Projectiles with air resistance uses particle kinetics.

Particle kinetics can be used to analyze objects following projectile motion with air resistance.

Did you know?

If we have objects which are either very massive, very small, or moving very fast, then Newton's equations do not provide a good model of their motion. Instead we must use Einstein's equations of general relativity (for massive and fast objects) or the equations of quantum mechanics (for very small objects). Unfortunately, these two theories cannot be used together, so we currently have no good models for objects which are simultaneously very small and very massive, such as micro black holes or the universe shortly after the big bang. Physicists are currently trying to reconcile general relativity with quantum mechanics by devising a new set of equations (sometimes called quantum gravity or a theory of everything). Current possibilities for new equations include string theory and loop quantum gravity, but none of these are generally accepted yet.

It is important to remember that all of these different equations are only models of reality and are not actually real:

“All models are wrong. Some models are useful.”
— George Box

Method of assumed forces and method of assumed motion

Newton's equations can be used in two main ways. Either we know the forces and we use this to compute the acceleration of a mass, or we know the acceleration and use this to compute the forces.

$$ \begin{align}\textbf{Method of assumed forces: } \text{Know } \vec{F} \Longrightarrow \text{Compute } \vec{a} \\ \textbf{Method of assumed motion: } \text{Know } \vec{a} \Longrightarrow \text{Compute } \vec{F} \end{align} $$

Of course there are other possibilities from these two, such as knowing the vertical component of the force and the horizontal component of acceleration, and then computing the missing components of each.

Example Problem: Method of assumed forces #rep-xf

A cannonball of mass $ m $ is fired from the origin with initial velocity $ \vec{v}_0 $ and then experiences a force $ \vec{F}_{\rm g} = -m g \hat{\jmath} $ due to gravity and a wind force $ \vec{F}_{\rm w} = -C_{\rm w} \hat{\imath} $. What is the motion of the cannonball as a function of time?

We can use Newton's equations to find the acceleration of the cannonball, using the method of assumed forces:

$$ \begin{aligned} m \vec{a} &= \vec{F}_{\rm w} + \vec{F}_{\rm g} \\ \vec{a} &= \frac{1}{m} \left( - C_{\rm w} \hat{\imath} - m g \hat{\jmath} \right) \end{aligned} $$

Example Problem: Method of assumed motion #rep-xm

A car of mass $ m $ is observed driving on a sinusoidal road at a constant horizontal speed $ v_0 $. The road surface has equation $ y = A \sin(k x) $, where $ A $ is the amplitude and $ k $ is the wavenumber. What is the force of the road on the car? Gravity $ g $ acts vertically and assume no air resistance.

We can determine the acceleration of the car, and then use Newton's equations and method of assumed motion to find the total force and thus the road force. First, we find the acceleration:

$$ \begin{aligned} x(t) &= x_0 + v_0 t \\ \vec{r}(t) &= x(t)\,\hat{\imath} + y(x(t)) \,\hat{\jmath} \\ &= (x_0 + v_0 t)\,\hat{\imath} + A \sin(k x_0 + k v_0 t)\,\hat{\jmath} \\ \vec{a}(t) = \ddot{\vec{r}}(t) &= - A (k v_0)^2 \sin(k x_0 + k v_0 t)\,\hat{\jmath} \\ &= - A (k v_0)^2 \sin(k x)\,\hat{\jmath}. \end{aligned} $$

Given the force of gravity $ \vec{F}_{\rm g} = - m g \,\hat{\jmath} $ and the road force $ \vec{F}_{\rm r} $, Newton's equations give the forces as:

$$ \begin{aligned} \vec{F}_{\rm r} + \vec{F}_{\rm g} &= m \vec{a} \\ \vec{F}_{\rm r} &= - m A (k v_0)^2 \sin(k x)\,\hat{\jmath} + m g \,\hat{\jmath} \\ &= m \Big(g - A (k v_0)^2 \sin(k x)\Big) \,\hat{\jmath}, \end{aligned} $$

where we have solved for the road force on the car.

Solution steps

The steps involved in analyzing a mechanical system with Newton's equations are as follows.

$$ \begin{aligned} &\text{1. FBD: draw a Free Body Diagram.} \\ &\text{2. Kinematics: determine $\vec{a}$.} \\ &\text{3. Newton: use $\vec{F} = m\vec{a}$.} \\ &\text{4. Algebra: rear and solve as needed.} \end{aligned} $$

Example Problem: Pendulum with Newton's equations #rep‑xl

Consider the 2D pendulum with a massless rigid rod of length $ \ell $ and a point mass $ m $. What is the equation of motion for $ \ddot\theta $ and the tension $ T $ in the rod?

It is helpful in this problem to use both Cartesian and polar bases:

1. FBD: The free body diagram for the point mass is:

The forces on the free body diagram are:

$$ \begin{aligned} \vec{F}_g &= - mg \,\hat\jmath \\ \vec{F}_T &= - T \,\hat{e}_r. \end{aligned} $$

2. Kinematics: Using the polar basis acceleration equation #rkv-ep gives:

$$ \begin{aligned} \vec{a} &= (\ddot{r} - r\dot\theta^2) \,\hat{e}_r + (r\ddot\theta + 2\dot{r}\dot\theta) \,\hat{e}_\theta \\ &= -\ell \dot\theta^2 \,\hat{e}_r + \ell\ddot\theta \,\hat{e}_\theta. \end{aligned} $$

3. Newton: Using #rep-en gives:

$$ \begin{aligned} \vec{F} &= m\vec{a} \\ \vec{F}_T + \vec{F}_g &= m(-\ell \dot\theta^2 \,\hat{e}_r + \ell\ddot\theta \,\hat{e}_\theta) \\ -T\,\hat{e}_r - mg\,\hat\jmath &= -m\ell \dot\theta^2 \,\hat{e}_r + m\ell\ddot\theta \,\hat{e}_\theta. \end{aligned} $$

4. Algebra: To compare components in the above equation we need to switch to a single basis. We will convert to $ \hat{e}_r,\hat{e}_\theta $ using:

$$ \hat\jmath = -\cos\theta \,\hat{e}_r + \sin\theta \,\hat{e}_\theta, $$

which gives:

$$ (-T + mg\cos\theta)\,\hat{e}_r - mg\sin\theta\,\hat{e}_\theta = -m\ell \dot\theta^2 \,\hat{e}_r + m\ell\ddot\theta \,\hat{e}_\theta. $$

Equating the $ \hat{e}_\theta $ and $ \hat{e}_r $ terms gives $ \ddot\theta $ and $ T $ by:

$$ \begin{aligned} \ddot\theta &= - \frac{g}{\ell} \sin\theta \\ T &= mg\cos\theta + m\ell\dot\theta^2. \end{aligned} $$

A free-body diagram (abbreviated as FBD, also called force diagram) is a diagram used to show the magnitude and direction of all applied forces, moments, and reaction and constraint forces acting on a body. They are important and necessary in solving complex problems in mechanics.

What is and is not included in a free-body diagram is important. Every free-body diagram should have the following:

The body represented as a dot if it is a point mass, and the body itself if it is a rigid body.
The external forces/moments. The force vector should indicate: relative magnitude, point of application, and the direction.
A properly defined coordinate system

A free-body diagram should not include the following:

Bodies other than the body we are interested in.
Forces applied by the body
Internal forces depending on the chosen system. For example, a free-body diagram on a truss should not include the forces between individual truss members.
Kinematic quantities (velocity and acceleration).

Warning!

Always assume the direction of forces/moments to be positive according to the appropriate coordinate system. The calculations from Newton/Euler equations will provide you with the correct direction of those forces/moments. Things that should not follow this are:

Gravity
Tension
Friction if the velocity $ \vec{v} $ is provided

Warning!

If forces/moments are present, always begin with a free-body diagram. Do not write down equations before drawing the FBD as those are often simple kinematic equations, or Newton/Euler equations.

Numerical integration

Independent variable (time).
State variables come in pairs (position, velocity)($ \theta , \omega $).
Initial conditions are the state variables at $ t=0 $.

$$ \theta_0 = \theta(0) , \omega_0 = \omega(0) $$
Time step $ \Delta t $ is how we jump forward in time.
Update rule.

$$ \theta_{n+1} = \theta_n + \Delta t \dot{\theta_n} = \theta_n + \Delta t \omega_n $$

$$ \omega_{n+1} = \omega_n + \Delta t \ddot{\theta_n} = \omega_n + \Delta t \alpha_n $$
Need to compute second derivatives ($ \ddot{\theta}=\alpha $ ) at each timestep.