CHAPTER VIII.

TANGENTIAL AND NORMAL ACCELERATIONS.

136. IF mT and mN be the forces acting on a particle of mass m in directions of the tangent and normal, the equations of motion are

and

ms=mT, or mv. =mT,

dv

ds

We have already made a slight use of the expression for normal acceleration in article (103); we now proceed to develope, somewhat at length, the utilization of these expressions.

Motion of a heavy particle on a smooth curve in a vertical plane.

Measuring a horizontally, and y vertically downwards, and taking R as the normal reaction of the curve, measured outwards, the equations of motion are

†m (v2 — u2) = mg (y — y'),

if u be the velocity when y = y.

This is, in effect, the equation of energy, and can be written down at once from the assumption of the truth of the principle of energy.

The second equation determines the pressure.

137. Motion of a particle on a smooth curve under the action of forces to fixed centres, the forces being functions of the distances from those centres.

If r, r',... be the distances of the particle from the centres of force, and mP, mP'... the forces, the equations of motion are

this, which is the equation of energy, gives the velocity and the second equation determines the pressure.

138. Motion of a heavy particle, placed on the outside of a smooth circle and allowed to slide down.

If the particle start from the point Q, at an angular distance a from the vertex, and v be the velocity at P,

mg cos 0 R, R being the outward reaction of

.. R= mg (3 cos 0 – 2 cos a),

shewing that the pressure vanishes, and that the particle flies off the curve, when

cos 0 = cos a.

139. Motion of a heavy particle inside a smooth circular tube in a vertical plane.

We shall suppose that the particle starts with a given velocity u from the lowest point B.

R being the pressure, inwards, of the tube on the particle.

To find the height of ascent we put v=0, and to find the point where the pressure vanishes we put R=0.

(1) Take u2 <2ga; then the highest point is given by 2ga cos 0= 2ga - u2, and the pressure never vanishes.

(2) If u2 = 2ga, the particle rises to C and the pressure

then vanishes.

(3) If u2 > 2ga and <4ga, the highest point is given by

and the pressure vanishes, and changes sign, when

(4) If u2 > 4ga and <5ga, the particle rises to A and passes over and the pressure vanishes when

(5) If u2= 5ga, the pressure vanishes at A.

(6) If u2 > 5ga, the pressure never changes sign.

Oscillation of a Pendulum.

140. A heavy particle, suspended by a weightless string from a fixed point, and oscillating in a vertical plane, forms a simple pendulum.

Measuring from the vertical, and observing that if a be the length of the string, s=a0, the equation of motion is

If the amplitude of oscillation be very small, the approximate equation is

This represents isochronous vibrations, the time of a

complete vibration being 2π

Finite motion of a Pendulum.

Recurring to the equation ag sin 0, and supposing the pendulum to start at an inclination a to the vertical, we obtain

T

a2 = 2g (cos - cos a),

and therefore, if be the time of oscillation from one side to the other,

If Ø be the angle at the time t from the lowest point,