dynamics notes - complete

ENGG 1210 Dynamics Pg. 1

10. Dynamics

We will now begin the study of dynamics, or the analysis of bodies in motion.

Dynamics includes:

Kinematics: study of the geometry of motion. Kinematics is used to relate displacement,

velocity, acceleration, and time without reference to the cause of motion.

Kinetics: study of the relations existing between the forces acting on a body, the mass of

the body, and the motion of the body. Kinetics is used to predict the motion caused by

given forces or to determine the forces required to produce a given motion.

First, we will look at the dynamics of particles, meaning that the motion of the object or

particle will be considered without regard to its size. This essentially means that motion

or rotation of the object about its own centre will be neglected.

• Rectilinear motion: position, velocity, and acceleration of a particle as it moves

along a straight line.

• Curvilinear motion: position, velocity, and acceleration of a particle as it moves

along a curved line in two or three dimensions.

Position – a distance, or coordinate in space, defining the location of an object at a

certain time.

Units: m

Can be defined with respect to time, i.e. x = 6t2-t

3

Velocity – distance travelled in a certain amount of time, magnitude of the velocity is the

speed, velocity is a vector and therefore has a direction, magnitude and sense.

Units: m/s

Positive velocity indicates that the particle is moving in the positive direction, negative –

particle is moving in the negative direction.

t

xvelocityaverage

dt

dxv

Acceleration – rate of change of velocity; acceleration is also a vector quantity

Units: m/s2

Positive acceleration indicates that velocity is increasing, either by the particle moving

faster in the positive direction, or slower in the negative direction.

t

vonacceleratiaverage

dt

dva

dx

dvva


Particle motion defined by x = 6t2-t

3

2312 ttdt

dxv

tdt

dva 612

Particle starts with 0 velocity, but

positive acceleration at time t=0;

At t=2s, acceleration is zero, velocity

is at a maximum;

From t=2 to t=4s, v is positive, a is

negative, particle still moves in the

positive direction, just more and more

slowly (decelerating);

At t = 4s, velocity is 0 and position has

reached the maximum;

After t = 4s, v and a are both negative,

particle is accelerating in the negative

direction

At t = 6s, particle passes through the

origin, total distance travelled is 64 m.

Particle continues to move in the

negative direction.


For particle in uniform rectilinear motion, the acceleration is zero and the velocity is

constant.

vtxx

vtxx

dtvdx

vdt

dx

tx

x

0

0

00

constant

For a particle in uniformly accelerated rectilinear motion, the acceleration is constant, and

the following equations can be derived:

adt

dvconstant

tv

v

dtadv00

atvvatvv 00 ; - equation 1

atvdt

dx0

dtatvdxtx

x 0

0

0

2

002

1attvxx - equation 2

Or adx

dvv constant

x

x

v

v

dxavdv00

0

2

0

2

2

1xxavv

0

2

0

2 2 xxavv - equation 3

These equations are useful for objects in free-fall due to gravity.


Example question #1

The cheetah can run as fast as 75 mi/h. Assume that the animal’s acceleration is constant

and that it reaches top speed in 4 s, what distance can it cover in 10 s?


10.1 Motion of Several Particles

10.1.1 Relative motion of Two Particles

Consider two particles moving along the same straight line. If the position of A and B

are measured with respect to the same origin, the difference xB-xA defines the relative

position coordinate of B with respect to A (xB/A).

ABAB xxx / ; ABAB vvv / ; ABAB aaa /

10.1.2 Dependent Motion

O A B

xA xB/A

xB


Sometimes, the position of a particle may depend on the position of one or more other

particles. In the figure above, the position of block B depends on position of block A.

Since rope is of constant length, it follows that sum of lengths of segments must be

constant.

xA + 2xB = constant

Only one of xA or xB can be chosen arbitrarily – this system has 1 degree of freedom

(DOF)

Two coordinates can be chosen arbitrarily – this system has 2 DOF

In the above figure, the positions of three blocks are dependent.

2xA + 2xB + xC = constant

For linearly related positions, similar relations hold between velocities and

accelerations.

2vA + 2vB + vC = constant

2aA + 2aB + aC = constant


10.2 Curvilinear Motion

We have now seen motion along a straight line path, which can be defined by scalar

values. But motion along a curved path requires description using vectors. This is most

easily done using Cartesian vectors.

Consider a point P (x,y,z) whose position is given by the Cartesian vector r

:


Example Question #1

The skier leaves the 20° surface at 10 m/s. Determine the distance d to the point where

he lands, and the magnitudes of his velocity parallel and perpendicular to the surface just

before he lands.


10.3 Normal and Tangential Components

Consider the motion of a particle along a curved path. The figure below shows the

velocity of the particle at times t and t + ∆t. The change in velocity ∆v has two

components:


11. Dynamics - Kinetics

11.1 Newton’s Second Law

dt

vdmamF

(No longer in static equilibrium where sum of the forces =0, now dynamic equilibrium.)

For an object of mass m, moving in a Cartesian coordinate system:

kajaiamkFjFiF zyxzyx

Example Problem #4

The two crates shown are released from rest. Their masses are mA = 40 kg and mB = 30

kg, and the coefficients of friction between crate A and the inclined surface are s =0.2

and k =0.15. What is the acceleration of the crates?


11.2 Normal and Tangential Components

We have seen how to resolve an object’s acceleration in terms of a normal and tangential

component. Therefore, we can also write Newton’s second law in the following form:

)( nnttnntt eaeameFeF


Example Problem #5

A civil engineer’s preliminary design for a freeway off-ramp is circular with radius R =

60 m. If she assumes that the coefficient of static friction between the tires and the road

is at least s =0.4, what is the maximum speed at which the vehicles can enter the ramp

without losing traction?

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