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 train enters a curved horizontal section of track at a speed of 100 km/h and slows down
with constant deceleration to 50 km/h in 12 seconds. An accelerometer mounted inside the
train records a horizontal acceleration of 2 m/s2 when the train is 6 seconds into the curve.
Calculate the radius of curvature ρ of the track for this instant.
50 km/h at t=12 s
2 m/s2 at t=6 s
100 km/h at t=0
when t=6 s, ρ=?
vt =12 = vo + at t
⇒
50 100
=
+ at (12 − 0 )
3.6 3.6
⇒
vt = 6 =
at = −1.157 m / s 2
vt = 6 = v o + a t t
100
− 1.157(6 − 0 )
3.6
vt = 6 = 20.836 m / s = 75 km / h
a = at2 + a n2 ⇒
2=
(− 1.157 )2 + an2
a n = 1.63 m / s 2
an =
v2
ρ
⇒
v 2 20.836 2
ρ=
=
= 266.12 m
1.63
an
3. The design of a camshaft-drive
system of a four-cylinder automobile
engine is shown. As the engine is
revved up, the belt speed v changes
uniformly from 3 m/s to 6 m/s over a
two-second interval. Calculate the
magnitudes of the accelerations of
points P1 and P2 halfway through
this time interval.
Calculate the magnitudes of the accelerations of
points P1 and P2 halfway through this time interval.
P1 : circular motion
P2 : rectilinear motion
v 2 = v o + at t ⇒
6 = 3 + at (2 )
at = 1.5 m / s 2
v1 = vo + at t ⇒
v1 = 3 + 1.5(1)
v1 = 4.5 m / s (velocity halfway through this time interval – at t=1 s)
a P1 = at2 + a n2
a P1 = 337.5 m / s 2
a P2 = at = 1.5 m / s 2
⇒
2

2  4.5 
a P1 = (1.5) +
 0.06 


2
4. A small particle P starts from point O with a negligible speed and increases its
speed to a value v = 2 gh , where y is the vertical drop from O. When x=15 m,
determine the n-component of acceleration of the particle.
y = 0.008 x 2
x = 15 m
⇒
an = ?
x = 15 m , y = 0.008 x 2 = 1.8 m
v = 2(9.81)(1.8) = 5.94 m / s
an =
  dy  2 
1 +   
  dx  
ρ=
d2y
2 3/ 2
0.016
ρ
3/ 2
dy
= 0.016 x ⇒
dx
dy
= 0.016(15) = 0.24
dx x =15
dx 2
[1 + 0.24 ]
ρ=
v2
= 67.98 m ⇒
5.94 2
= 0.52 m / s 2
an =
67.98
,
d2y
dx
2
= 0.016
5. A small projectile is fired from point A with an initial velocity u=500 m/s at
the angle of 60o from the horizontal as shown. Neglect atmospheric resistance
and any change in g and compute the radius of curvature ρ of the path of the
projectile and the time rate of change of the speed 30 s after the firing.
u=500 m/s
6. 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 ρ of the path
followed by P.
7. 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 ρ 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.
8. In the design of a control mechanism, the vertical slotted guide is moving with a
constant velocity x =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 ρ of the path at this position.