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CES Mechanics II Lecture 2

RWTH Aachen University, Bachelor program in Computational Engineering Science

Topics:

• Force-mass-acceleration method

Although we will not fully focus on forces until we study kinetics later in the semester, it is important to realize that accelerations in kinematics are directly related to forces acting on a particle. A standard way of analyzing a motion of particles is the direct method (known as foce-mass-acceleration method) where forces on a particle are computed and equated to mass times acceleration according to second law of Newton (\mathbf{F} = m \mathbf{a}).

In statics, we have used free-body diagrams as a help with writing down equilibrium equations. Now, we can either

• continue to use FBD only and treat the "inertia" m \mathbf{a} as a pseudo-force that is added to real forces with negative sign, to help us write \mathbf{F} - m \mathbf{a} = \mathbf{0}, or
• draw regular FBD, draw a mass-acceleration diagram (MAD) identifying m \mathbf{a} next to it (e.g., identifying known component or known direction of this vector), and equate the two to obtain the equation \mathbf{F} = m \mathbf{a}.
• Rectilinear motion (geradlinige Bewegung)

When forces act in one direction only, we have rectilinear motion. This can occur naturally, or due to path constraints. We will assume that x is the motion coordinate, so \sum F_y = \sum F_z = 0. We will drop the dimension subscripts and write a = F / m and v as acceleration and velocity along the direction of motion.

• Acceleration, velocity, displacement in rectilinear motion 1.1.3

In general, resultant force and thus acceleration can be a function of velocity, position and time: a = a(v,x,t). This can be also written as \ddot{x} = f(\dot{x}, x, t), which is known as an ordinary differential equation (ODE, gewöhnliche Differentialgleichung). You will learn how to solve ODEs in later semesters; for now we will focus on special cases in which ODE knowledge is not necessary.

• Case 0: Constant acceleration

This case arises in constant gravitational field, or constant electric field. When a = \mathrm{const}, velocity can be easily obtained by integration with respect to time between initial time t_0 at which velocity is known to be v_0 and current time t:

v(t) = v_0 + \int_{t_0}^t a d\bar{t} = v_0 + a \int_{t_0}^t d\bar{t} = v_0 + a (t - t_0),

where the variable of integration is denoted as \bar{t} to distinguish it from the current time t.

Similarly, position can be obtained by another integration, assuming that at t_0 the position is known to be x_0:

x(t) = x_0 + \int_{t_0}^t v(t) d\bar{t} = x_0 + \int_{t_0}^t \left( v_0 + a (\bar{t} - t_0) \right) d\bar{t} = x_0 + v_0 (t - t_0) + a \frac{(t - t_0)^2}{2}.

In the common case when t_0 = 0, the equations simplify to those seen in high school physics class:

v(t) = v_0 + a t,

x(t) = x_0 + v_0 t + \frac{a t^2}{2}.

• Case 1: Acceleration is a function of time

When a = a(t), velocity can be obtained in most cases by straightforward integration:

v(t) = v_0 + \int_{t_0}^t a(\bar{t}) d\bar{t},

or, using indefinite integral notation:

v(t) = \int a(t) dt + C_1,

where C_1 is an arbitrary constant of integration.

Similarly, position can be obtained by another integration:

x(t) = x_0 + \int_{t_0}^t v(\bar{t}) d\bar{t},

or, using indefinite integral notation:

x(t) = \int v(t) dt + C_2,

where C_2 is an arbitrary constant of integration.

• Case 2: Acceleration is a function of position

This case arises when the forces depend on position, for example, in case of spring (elastic) forces or gravitational force acting over long distance. When a = a(x), a formula for velocity as a function of position can be obtained by separation of variables:

\frac{v(t)^2}{2} = \frac{v_0^2}{2} + \int_{x_0}^x a(\bar{x}) d\bar{x},

or, using indefinite integral notation:

\frac{v(t)^2}{2} = \int a(x) dx + C_3,

where C_3 is an arbitrary constant of integration.

The next formula can be used to derive time as a function of position:

t(x) = t_0 + \int_{x_0}^x \frac{d\bar{x}}{v(\bar{x})},

or, using indefinite integral notation:

t(x) = \int \frac{dx}{v(x)} + C_4,

where C_4 is an arbitrary constant of integration.

The expression for t(x) can be often inverted to obtain x(t), and then v(t) and a(t) by simple differentiation. In the next lecture, we will discuss Case 3.

Corresponding Gross et al. Kinetik chapters are shown in blue.

Key techniques:

• drawing FBD and Mass-Acceleration Diagram (MAD) or drawing the inertia vector with opposite sign on the FBD
• relating force to acceleration
• deriving velocity and displacement from acceleration
• drawing and interpreting the velocity-time, displacement-time, and velocity-displacement diagrams

ConcepTests (RWTH only):