Light waves are electromagnetic waves that travel at (a) Find the frequency of t

Question

Light waves are electromagnetic waves that travel at

(a) Find the frequency of this light wave.
Hz

(b) Find its period.

The period of motion of an object-spring system is T = 0.534 s when an object of mass m = 266 g is attached to the spring.

(b) Find the force constant of the spring.

(c) If the total energy of the oscillating motion is 0.189 J, find the amplitude of the oscillations.

A horizontal block-spring system with the block on a frictionless surface has total mechanical energy E = 41.4 J and a maximum displacement from equilibrium of 0.265 m.

(d) What is the speed of the block when its displacement is 0.160 m?
m/s

(e) Find the kinetic energy of the block at x = 0.160 m

(f) Find the potential energy stored in the spring when x = 0.160 m.

(g) Suppose the same system is released from rest at x = 0.265 m on a rough surface so that it loses 15.0 J by the time it reaches its first turning point (after passing equilibrium at x = 0). What is its position at that instant?

Explanation / Answer

a)

velocity = frequency times wavelength
f = (3 X 10^8 m/s)/(5.5 X 10^-7 m) = 5.45 X10 ^14 Hz
b)

Period = 1/f
T = 1 / (5.45 X10 ^14 Hz) =1.8 X 10^-15 s

b)
T = 2pi?(m/k)
0.534^2 = 4pi^2*m/k
k = 4pi^2*0.266/0.534^2
k = 36.789 N/m

c)
0.189 J = 1/2*kA^2 (total energy = spring energy when the displacement is maximum = amplitude. )
A^2 = 0.189*2/36.789
A= 0.01027 m = 1.027 cm > amplitude

D)E = (1/2)*M*vmax^2
M = 2*E/vmax^2
= 2 * 41.4/3.45^2
= 6.9565 kg

E = 1/2 *k*A^2
k = 2E/A^2
= 2*41.4/0.265^2
= 1179.06 N/m  

E = (1/2)Mv^2 + (1/2)kx^2
41.4 = (1/2)*6.9565*v^2 + (1/2)*1179.06*0.16^2
41.4 = 3.47825 v^2 + 15.0919
3.47825 v^2 = 26.3080
v = sqrt(26.3080/3.47825) = 2.75 m/s
Ans: 2.75 m/s

E) KE = (1/2)Mv^2 = (1/2)*6.9565*2.75^2 = 26.304 J
Ans: 26.304 J

F) PE = E-KE = 41.4 - 26.304 = 25 J
Ans: 15.0957 J

G) PE1 = (1/2)*k*x1^2 = (1/2) * 1179.06 * 0.265^2 = 41.4 J
KE1 = 0
KE2 = 0
x2 = ?
Loss of energy = 15 J
Therefore KE2 + PE2 = KE1 + PE1 - 15
0 + PE2 = 0 + 41.4 - 15
PE2 = 26.4 J
(1/2)*k*x2^2 = 26.4
(1/2) * 1179.06 * x2^2 = 26.4
589.53 x2^2 = 26.4
x2 = sqrt(26.4/589.53) = 0.2116 m
Ans: 0.2116 m

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