High School (hard)

Challenge 7 – Music, maestro

The length of the string of a certain guitar is 60cm. Its fundamental frequency is 247Hz.

a) What is the speed of transverse waves on the string?

b) If the linear mass density of the string is 0.01g/cm What should be its tension when tuned?

Challenge 6 –  Different fingerboards

Two tuning forks are struck simultaneously and we can hear four beats per second. The frequency of a tuning fork is 500Hz.

a) What are the possible values of the pitch frequency of the other fingerboard?

b) A piece of wax is placed in the range from 500Hz to reduce slightly their frequency. Explain how you can use the measure of the new beat frequency to determine which of the answers to part a) is the correct frequency of the second pitch.

Challenge 5 – Calculation of water outlet

There is a cylindrical tank filled with water to a height H. It also has a faucet at a depth h (in water). Consider that the hole of the faucet is proportionally much smaller than the area of the cylinder base. Then:

a) Find the distance x which the water impinges on the ground as a function of h and H.

b) Show that there are two values of h that are equidistant from the point h = 1/2H that give the same distance x.

c) Show that x is maximal when h = 1/2H What is the value of this maximum distance x?

Challenge 4 – Half cylinder

We have a half cylinder (see figure) of mass M and radius R that rests on a horizontal surface, with the flat side facing up. If one side of the half cylinder is pushed slightly and then released, the object will oscillate around its equilibrium position. Determine the period of oscillation.

How does it changes in the case of a hemi-sphere?

 

Challenge 3 – Uniform linear motion in an inertial system

In an inertial system, a body moves freely with a trajectory given by:

x = v · t, y = 0, z = 0

What is the trajectory in a rotating system with constant angular velocity ω counterclockwise around the z axis?

Challenge 2 – Lenses and mirrors

Lateral amplification of a spherical mirror or a thin lens is given by m =-s ‘/ s, where s is the distance of the lens to the object and s’ is the distance of the object in the image. Demonstrate that in the case of small horizontal extension objects, the longitudinal amplification is approximately -m².

Hint: Show that ds ‘/ ds = s’ ² / s ².

Challenge 1 – The Atwood’s Machine

The apparatus shown in the image is known as the Atwood’s machine.

It is used to measure acceleration due to gravity g from the acceleration of the blocks (the blue and red squares).

Assuming that the rope and pulley have negligible mass with respect to the blocks and there is no friction,

a) Calculate the acceleration of the blocks and the string tension T as a function of the masses of both blocks and the constant g.

The determination of the gravitational constant g can be determined by measuring the time t it takes for a mass to travel a distance L from rest.

b) Determine an expression for g as a function of the masses  m₁, m₂, L y t.

c) Demonstrate that if a small mistake is made in the measurement of time dt, it will lead to an error in the determination of g, given by the expression dg/g=-2dt/t.

d) If L = 3 m and m₁=1kg  determine the value of  m₂ so that g can be measured to an accuracy of 5% and a time measurement error of less than 0.1 s.

Note: Assume that the only significant uncertainty is time.