Some questions on Simple harmonic motion, please help!?

Question

7) Prakriti’s experiment involves an object (hexagonal nut, m = 2.2 g) attached to the mid-point of a light string. The length of string from the

vibration generator (left) to the pulley (right)

is L = 40 cm. Tension in the string

is provided by a 200 g weight.

a) Find the tension in the string.

b) Find an expression for the net vertical force pulling on the object when vertical displacement is y. (Ignore gravitational force on the nut; tension is much larger)

The string pulls on both sides of the nut. Use the lengths in the figure to express sinθ.

c) Find the natural frequency of this system.

10) In question 7 you determined the natural frequency of the “loaded string”, which can be observed if the mass is pulled as shown in the figure, and then released.

a) Describe what happens to the amplitude of successive oscillations.

b) What is the name of this phenomenon, and where does the energy go?

The left end of the string is attached to a vibration generator which, if turned on, makes one end of the string move up and down.

c) What can be observed when the frequency of the generator (driving frequency) is much higher than the natural frequency of the system?

d) What can be observed when the driving frequency approaches the natural frequency of the system? What is the name of this phenomenon?

e) What can be observed when the driving frequency is very low compared to the natural frequency of the system?

f) What changes would occur if the string and the object were submerged in water?

11) You are on a shopping spree, taking your 21-gear bicycle to the grocery shop. You head home with one full shopping bag on each handle bar, carefully balancing through the traffic. Everything goes well until you reach a climb through the outer gardens of your estate, where you must pedal hard. Suddenly, the bags start swinging violently, and you are about to lose control of the bike.

a) Give a physical explanation of what happens.

b) How can you keep up both appearances and speed? (i.e. how can you solve the problem without stopping or

Thankyou very much if you answer this. It will be really gratefull and helpfull!

Answer 1

(7) It would be easier for us if you posted a sketch of the experiment. If I interpret it right, the problem can be restated as follow :

"A horizontal string of length L = 40 cm is attached to a vibrator at one end and a mass M= 200g at the other, thru a pulley. At the midpoint of the string is a nut of mass m= 2.2g. Evaluate the vertical force acting on the nut when it is t a distance y off its equilibrium position. Deduce the natural frequency."

SOLUTION

(7a) Tension of the string:

T = Mg =0.200×9.8 = 1.96 N

(7b) When the nut have a vertical displacement y meter, each of the halves of the string makes an angle α with the horizontal

tanα = y/(L/2) = y/0.20 = 5y

Each half of the string acts on the nut with a force of magnitude T in the direction α above the horizontal.

Horizontal components cancelling each other, the resulting force is vertical, upward, of magnitude

F = 2Tsinα = 2Ttanα /√(1 + tan²α)

F = (2×1.96×5y) / √(1 + 25y²)

F = 19.6 y / √(1 + 25y²)

(7c) When the displacement y is very small with respect to the length L of the string, for example, take y = 1 cm = 0.01 m, 25y² = 0.0025 << 1, we can neglect the term 25y² in the radicand. The magnitude of the force will be

F = 19.6 y

This force tends to pull the nut to its equilibrum position, its expression can be written as

F = -19.6 y = -ky with k = 19.6 N/m

The natural frequency of oscillation with small amplitudes of this system is

f = (1/2π) √(k/m)

f = (1/2π) √(19.6/0.0022) = 94.388 / 2π

f = 15 Hz