Tangential and Normal Accelerations. Examples 99 Therefore the acceleration along the tangent ^^~^ '
^„„
u.
'^^il
,
„:„ ^
dv
_ <^^s
_dvds _
dv
cm
and the acceleration along the normal = - --7 sin
+
cli-
| cos = — — dtp
a curve is described by a particle having a constant acceleration 90. -Ej;. in a direction inclined at a constant angle to the tangent; shew that the curve is an equiangular
spiral.
Here -3-=/ cos o and
1 ds .'.
jr
2 .: .-.
which
3-,
d^
— =/sin
o,
where /and a are constants,
=s cot a +^, where 4
is
a constant.
+ .4) = 2^ cot a + const. s= -^tano + jBe^'/'Cota log
is the intrinsic
(s
cot
tt
equation of an equiangular spiral.
EXAMPLES Find the intrinsic equation to a curve such that, when a point moves on it with constant tangential acceleration, the magnitudes of the tangential velocity and the normal acceleration are in a constant ratio. 1.
2.
A
point moves along the arc of a cycloid in such a manner that it rotates with constant angular velocity ; shew that the
the tangent at
acceleration of the
A
3.
moving point
accelerations are equal velocity 4.
is
constant in magnitude.
point moves in a curve so that
its tangential and normal and the tangent rotates with constant angular
find the path.
;
If the relation
between the velocity of a particle and the arc
it
has
described be
find the tangential force acting
on the particle and the time that must
elapse from the beginning of the motion 5.
till
the velocity has the value
V.
Shew that a
each point by generating
cycloid can be a free path for a particle acted on at a constant force parallel to the corresponding radius of the
circle, this circle
being placed at the vertex.
An
insect crawls at a constant rate u along the spoke of a cartwheel, of radius a, whose centre is moving in a straight line with velocity 6.
V.
Find 7.
its accelerations
A circle
rolls
along and perpendicular to the spoke.
on a straight
line,
instant being v and its acceleration
/;
the velocity of
its
centre at any
find the tangential
and normal
accelerations of a point on the edge of the circle who&e angi.dar distance from the point of contact is 6.
7—2