Rotation, Circle motion, Oscillation

Physical basics of Biophysics Dávid Szatmári

September 2014. Circle motion

Circular motion: when the path of an object is a circle the distance in this case is the arc (i).

The direction of this velocity vector is always perpendicular to the radius of a circle, which pointed out of circle its tangential, means we should call as tangential velocity.

The angle difference in radian determines the circle motion.

Δi = r ·Δα Periodical time: the requried time of motion to take one whole circle (T).

The average tangential velocity: v = Δi /Δt Angular velocity (scalar quantity): ω = Δα /Δt [1/s] v = r ω

Number of circles during a unit time is the revolution (n): n = 1/T; ω = 2π /T = 2πn;

However the velocity of body on circle path is changing by the time, it generates a special acceleration, acp - centripetal acceleration. Δv /Δs = v /r; Δv = Δs (v /r); Δv /Δt = Δs /Δt (v /r); 2 2 acp = v /r; acp = rω ; Harmonical circle motion

A motion is harmonical if the path is circle and the arcus of motion is proportional with the relapse time of motion. v, ω, acp = constant i = vt α = ωt

The acceleration of moving body is the centripetal acceleration, therefore the sum of affective forces are sowing in the centre of motion. The given force which enforce the body on a circle path is the centripetal force. 2 Fcp = m (v r) The dynamical criterium of harmonical motion. Constantly changing circle motion

Constantly changing circle motion of masspoint needs a resultant force which is built up by two components: - Tangential force : is changing the velocity along the path, its amount is constant, its direction is allways tangential. - Centripetal force : enforces the circle path of motion, its amount is proportional with the quadrat of the time, its direction parallel with the radius.

We can calculate the amount of resultant force by the Pythagorean equation: 2 2 Fr = √(F t + Fcp )

Accelerating motion The unit change in unit time of tangential velocity vector is the tangential acceleration. at = Δv /Δt

Angular acceleration: in case of constantly changing circle motion, the angular velocity 2 is propotional with the time. symbol: β [1/ s ] at = rβ ω = ω 0 + βt Rotational motion If the affective line of a given force does not cross the rotational axis of a rigid body, it rotates. The force initiated torque is proportional with the angular acceleration. M ~β

The ratio of them determines a constant, is the moment of inertia. symbol: Θ [kg m 2] M = Θβ Dynamical equation which describes the rotation.

Rotational energy: 2 Erot = ½ Θ ω A rotating body has a rotational quantity, is the angular momentum. symbol: N [(kg m 2)/s]

N = Θω M Δt = ΔN Law of Angular momentum: The change of angular momentum is equal with effect which initalized the rotation. M = 0; N = constant Angular momentum continuity: In a closed system if the rotating body is not effected by moment of inertia, the angular momentum is constant. Oscillation

• Any motion that repeats itself in equal intervals of time is called periodic or harmonic motion.

• If a particle in periodic motion moves back and forth over the same way, there is oscillatory or vibratory motion.

• A physical quantity which changes in the time periodicly. Equilibrium (balanced position): ΣF=0

Displacement ( x): the distance (linear or angular) of the oscillating particle from its equilibrium position.

Amplitude(A): the maximal displacement

Period ( T): is the required time for a whole round circle of the motion.

Frequency ( f): is the number of oscillations per unit of time.

Angular frequency (velocity) (ω): velocity of the circular motion.

Phase (ωt+α): the state of the object in given time.

Phase constant (α): describe the phase Harmonical oscillation

Displacement: x(t)= A sin(ωt+α) ω = 2π/T = 2πf, f = 1/T Velocity: v(t)= Aω cos(ωt+α) Acceleration: a(t)= Aω2 sin(ωt+α)

Force: F= k ∆x (k: spring constant)

The resultant oscillation as the sum of two independent oscillation.

x(t)= x 1(t)+ x 2(t)

Phase diffrence: δ = α2α1 Lissajous curves Attenuated harmonic oscillation

An outer force dissipates the energy of oscillation.

x=Ae βtsin (ωt+α) β: coeff. of attenuation e: special constant of the nature (2,718...)

Ratio of attenuation: K=A 1/A 2 Resonance

Resonance : If the oscillation is enhanced by an outer force. And if the frequency of outer force is equal with the proper frequency of the system. Resonance catastrophe Problems 1. Let determine the rotational energy of a helicopter, if the moment of inertia is 5 (kgm 2)/s 2, angular acceleration is 2 1/s 2 , and angular velocity is 0.25 1/s! E=0,156 J

1. Let determine the velocity and acceleration of a mass, which oscillates on a spring. It started from the equilibrum state, the elapsed time is 10 min, and the period is 2 s, the maximal stretched length of the spring is 0.1m (rad )! a=0, v=0,314 m/s