Classical Mechanics Answers

Let us consider a swing as a harmonic oscillator! The velocity of the swing at the lowest position is 3 m/s. The acceleration at the highest point is 3 m/s2 . Write down the displacement-time function!

The parameters of the spring oscillator shown by the figure are: m = 3 kg and k = 300 N/m. We displace the ball to the left by s = 10 cm from the equilibrium position, then release it. Let us assume there is no loss of energy in the system! Calculate:

a) the natural frequency of the system,

b) the period of the oscillation,

c) the amplitude of the oscillation! 

Waves are propagating on the surface of water towards the shore with a velocity of 1.5 m/s. The distance between two neighboring crests is six meters. There is a piece of wood somewhere further in the water that turns up and disappears periodically as the water waves when you are looking at it from the shore. Calculate the time interval between two turn-ups.

A coil 20 cm long with a radius of 1 cm has a winding of 1000

turns of copper wire with a cross-sectional area of 1 mm² . The coil is

connected to an alternating current circuit with a frequency of 50 Hz.

What part of the impedance of the coil will be formed by (1) the

resistance, and (2) the inductive reactance?

Two capacitors with capacitances of C₁ =0.3 μF and C₂=0.2 μF are

connected in series to an alternating current circuit with a voltage of 300

V and a frequency of 50 Hz. Find: (1) the intensity of the current in the

circuit, (2) the potential drop across the first and second capacitors.

A magnetic flux through a stationary loop with a resistance R

varies during the time interval τ as Ф = at (τ — t). Find the

amount of heat generated in the loop during that time. The

inductance of the loop is to be neglected.

An element is first connected across an external resistance R₁=2 Ω, and

then across an external resistance R₂= 0.5 Ω. Find the e.m.f. of the

element and its internal resistance if in each of these cases the power

evolved in the external circuit is the same and equal to 2.54 W.

Two long rectilinear conductors are arranged parallel to each other at a

distance of 10 cm. Currents of I₁ =I₂ =5 A flow through the conductors in

opposite directions. Find the magnitude and direction of the magnetic

field intensity at a point 10 cm from each conductor.

An electric kettle containing 600 cm³ of water at 9⁰C and with a heater

coil resistance equal to 16 Ω was left connected to the mains. In how

much time will all the water in the kettle boil away? The voltage in the

mains is 120 V and the efficiency of the kettle 60%.

Find the Hamiltonian and write the canonical equations of a harmonic oscillator, if its Lagrangian is L=(x ̇ ² – ω²x²)