Simple pendulum

Physical background

Kinematics of the curvilinear motion

The velocity of a curvilinear motion is defined

5_eq1 (1)

where r is the position vector. The direction of the velocity v is given by the tangent to the trajectory ( Figure 1).

The basic property of any curvilinear motion (even those with constant velocity) is its non zero normal acceleration an . It is due to the fact that the direction of the unit vector of the velocity is not constant and changes with time.

5_eq2 (2)

Its value is

5_eq3 (3)

Where v is the value of the instantaneous velocity and R is the radius of the curvature of the trajectory in and its direction is the normal to the trajectory.

If the motion has variable velocity, then we introduce, similarly as with linear motion, the tangential acceleration

5_eq4 (4)

With the direction given by the tangent to the trajectory.

5_tangencialne_normalove

Figure 1 Tangential and normal acceleration of the curvilinear motion (the angle of deflection in the figure is ?<0)

To study the curvilinear motion we will use the simple pendulum (Figure 1). It is a weight of the mass m, hanging on the suspension of the length r. The pendulum is moving under the influence of the force of weight G by the harmonic motion, described by the angular displacement ? (> < 0!)

5_eq5 (5)

where ?o is its amplitude and ? is the angular frequency 5_eq5awhere g is the acceleration due to the weight.

For the normal acceleration we can then write, using eq.(5), ( Figure 2)

5_eq6 (6)

where r is the radius vector value and 5_eq6a is the angular velocity of the pendulum motion.

For the tangential acceleration, using the angular acceleration 5_eq6b , we have

5_eq7 (7)

5_graf_tangencialne_normalove

Figure 2 Tangential and normal acceleration of the simple pendulum, eq.(6) and (7);

r = 2m, ?o= 0,1rad T = 3s, m = 0,1 kg

Dynamics of the curvilinear motion

Why is the simple pendulum moving according to eq. (5)? The equation of motion (2. Newton’ law of motion), expressing the action of forces from other bodies on the motion of a body is

5_eq8 (8)

where a is the acceleration of the body. For a simple pendulum from Figure3 the eq.(8) reads for the x axis (?

5_eq9 (9)

whereas for the y axis

5_eq10 (10)

Equation (9) is the differential equation for the angular displacement ? (where we used eq.(6))

5_eq11 (11)

The eq.(11) is analytically not soluble, but for small angular displacements (??) = ? and the solution is (see eq. (5))

5_eq11a

Then we can write, using eqs. (5) and (6) for the pull T in the suspension of the pendulum the important for the measurement expression ( see Figure 3 and its dependence in Figure 4 )

5_eq12 (12)

5_dynamika_pohybu

Figure 3 Dynamics of the simple pendulum motion

5_tah_zavesenia

Figure 4 Time dependence of the pull T in the suspension of the pendulum, eq.12;

r = 2m, ?o= 0,1rad T = 3s, m = 0,1 kg

Energy of the curvilinear motion

The kinetic energy of the pendulum is, using eq. (5) and the definition of the angular velocity

5_eq13

The potential gravitation energy of the weight of the pendulum is, using the expression for the pendulum position h above the reference level (taken as the lowest point of the pendulum swing) h = l(1-cos ?(t)) ( Figure 5)

5_eq14

For the pendulum motion the law of the conservation of the mechanical energy is fulfilled (Figure 6)

5_eq15 (15)