Simple Harmonic Motion: Problems?

Question

I have some question, can you please answer it by explaining.

thanks a lot in advance

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.

8) Imagine a mass m = 10 g oscillating in a spring (k = 10 Nm-1). Amplitude = 1 cm.

a) Describe how energy changes between potential energy and kinetic energy during one cycle of SHM. Where does the potential energy equal the total energy?

b) State the formula for potential energy stored in a spring, and show that this can be written as .

c) What is the total energy in this system?

d) What fraction is the kinetic energy of the total energy when the displacement is one half amplitude?

e) What fraction is the speed of the maximum speed when the displacement is one half amplitude?

9) You are in a spaceship, returning from Mars with rock samples. You want to determine the masses of the samples by attaching them to a spring and measuring the frequency of oscillation. How stiff must the spring be so that 1 kg corresponds to a frequency of 1 Hz? (Stiffness is expressed by the spring constant)

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

Kid! It’s too much to answer for one person! You’d better post 3 separate questions;

Sorry!

8) total energy of mass-spring system is E=E1+E2, where

E1=0.5*m*v^2 is kinetic energy, E2=0.5*k*x^2 is potential energy,

m=10gram=0.010kg, k=10N/m,

v is velocity of mass at deflection

x from equilibrium;

Since energy conserves E is a constant;

Therefore

the more deflection x the less velocity v;

the more velocity v the less deflection x;

max velocity is when x=0, and E(x=0)=E1 is kinetic energy wholly;

max deflection is when v=0, and

E(v=0)=E2= 0.5*k*Amplitude^2=

= 0.5*10*0.01^2 =5e-4 J is potential energy wholly;

(d) x=A/2, E2(x=A/2) =0.5*k*(A/2)^2 =1.25e-4 J, hence

E1=5e-4 –1.25e-4 =3.75e-4 J; thus E1/E = 3.75/5 =3/4;

(e) E1/E =(0.5*m*v^2)/(0.5*m*v0^2) = (v/v0)^2 =3/4, hence

v/v0 =0.5*sqrt(3);

9) frequency of a mass bouncing on a spring is f=sqrt(k/m), hence

k=m*f^2 =1*1^2 = 1 N/m;