One of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of the particles. t and n are two orthogonal axes considered separately for every instant of motion.

These coordinates provide a natural description for curvilinear motion and are frequently the most direct and convenient coordinates to use; they move along the path with the particle.

The positive direction for n at any position is always taken toward the center of curvature of the path. Therefore no position vector is required.

We now use the coordinates n and t to describe the velocity and acceleration. For this purpose, we introduce unit vectors in the n-direction and in the t-direction.

ne

te

During a differential increment of time dt, the particle moves a differential distance ds along the curve from A to A:

ds=r db

where b is in radians.

It is unnecessary to consider the differential change in r

between A and A for a differential time period.

brb

r dt

d

dt

dsv

tev

br

The magnitude of velocity:

br

bbb

r

v

dt

d

dt

d

dt

dsv

Therefore, the velocity in vector form is

tt eevv

br

The acceleration of the particle is defined as ,

and we observe that the acceleration is a vector which reflects both the change in magnitude and the change in direction of velocity.

dt

vda

We now differentiate the velocity applying the ordinary rule for the differentiation of the product of a scalar and a vector and get

tt

t evevdt

evd

dt

vda

The unit vector now has a nonzero derivative because although its magnitude stays the same, its direction changes. As the particle moves from point A to A, the unit vector becomes .

te

te

te

te

te

ted

ttt edee

The vectorial difference is:

To find we analyze the change in during a differential

increment of motion as the particle moves from A to A’.

te

te

In the limit, the magnitude of will

be . For a differential

time interval, the direction of can

be considered perpendicular to .

Therefore, the direction of can

be considered the same as that of .

ted

bb ddeed tt

te

ted

ted

ne

db

ted

te

te

ne

The derivative of will be obtained as, te

nt

nt ed

edoeded

bb r

By dividing to dt

nt

t

ntnt

ev

e

vvev

eeedt

d

dt

ed

r

rbbrbr

bb

,

nttt

ttnn

ev

eveveva

eaeaa

r

2

nttt

n

a

t

a

tt

ev

veveveva

veveveveva

nt

r

rbbrb

v

222

a ntnttt aaev

eveveva

r

Here,

22

22

2222

tn

t

n

n

aaa

sva

vv

a

vvv

a

br

bbrr

br

brbr

r

br

r

(Change in direction of the velocity)

(Change in magnitude of the velocity)

Radius of Curvature (Eğrilik Yarıçapı):

If trajectory is given as y=f(x), the radius of curvature can be determined with the following formula:

The absolute value sign is used to guarantee that r will always be positive “+”.

2

2

2/32

1

dx

yd

dx

dy

r

The change in velocity vector is and it indicates the direction of the acceleration vector . When is divided into two components one normal and the other tangent to the velocity, the normal component is defined as and in the limit it will be equal to the arc length obtained by rotating the velocity vector with magnitude v for an angle of db. The normal component of acceleration will be equal to the time derivative of | | .

vd

a

vd

bvdvd n

b

vdt

vda

n

n

nvd

nvd

The velocity vectors at A and A' are drawn to start at a same point in order to explain the change in the magnitude and direction of the veloicty vector and how these changes generate the normal and tangential components of the acceleration vector.

The tangential component of is . Its magnitude is equal to or the change in the length of the velocity vector. The tangential component of acceleration will be equal to the time derivative of

vd

tvd

tdvvv

tdv

svdt

vd

dt

dva

t

t

The normal component of acceleration an is always directed toward the center of curvature. The tangential component of acceleration at, on the other hand, will be in the positive t-direction of motion if speed v is increasing and in the negative t-direction if the speed is decreasing.

022

r

vv

an

At an inflection point on the curve, the normal acceleration goes to zero since r becomes infinite.

Figure shows the schematic representations of the variation in the acceleration vector for a particle moving from A to B with (a) increasing speed and (b) decreasing speed.

r = r = constant

rva

vrr

va

t

n

22

dt

d

dt

d

(Angular velocity)

(Angular acceleration)

Circular motion is an important special case of plane curvilinear motion where the radius of curvature r becomes the constant radius r of the circle and the angle b is replaced by the angle measured from any convenient radial reference to OP.

1. Six acceleration vectors are shown for the car whose velocity vector is directed forward. For each acceleration vector describe in words the instantaneous motion of the car.

2. A flywheel rotates with variable velocity. At a certain time, the tangential acceleration of point A on the flywheel is 1 m/s2, while the normal acceleration of point B on the flywheel is 0.6 m/s2. Determine the velocity of point A and the total acceleration of point B.

A

B 150 mm

200 mm

3. A baseball player releases a ball with the initial conditions shown in the figure. Determine the radius of curvature of the trajectory (a) just after release and (b) at the apex. For each case, compute the time rate of change of the speed.

4. A marble rolls down a chute which is bent in the shape of a parabola. Its equation is given as y=x2 – 6x + 9 [m] The marble passes point A with a speed of 3 m/s, which is increasing at the rate of 5 m/s2. Determine the normal and tangential components an and at, of the acceleration of the marble as it passes point A. Also determine the angle between the velocity vector and the acceleration vector at point A.

x (m)

y (m)

x=5 m

A

y=f(x)

5. The pin P is constrained to move in the slotted guides which move at right angles to one another. At the instant represented, A has a velocity to the right of 0.2 m/s which is decreasing at the rate of 0.75 m/s each second. At the same time, B is moving down with a velocity of 0.15 m/s which is decreasing at the rate of 0.5 m/s each second. For this instant, determine the radius of curvature r of the path followed by P.

6. A ball is thrown horizontally from the top of a 50 m cliff at A with a speed of 15 m/s and lands at point C. Because of strong horizontal wind the ball has a constant acceleration in the negative x-direction. Determine the radius of curvature r of the path of the ball at B where its trajectory makes an angle of 45o with the horizontal. Neglect any effect of air resistance in the vertical direction.

7. In the design of a control mechanism, the vertical slotted guide is moving with a constant velocity = 150 mm/s during the interval of its motion from x = -80 mm to x = +80 mm. For the instant when x = 60 mm, calculate the n- and t-components of acceleration of the pin P, which is confined to move in the parabolic groove. From these results, determine the radius of curvature r of the path at this position.

x