Science & Music

A chain hangs between two points in the same horizontal line. We want to compare the tension of the chain in 3 different points (A, B and C). Point C is lower than B and point B is lower than A. We can answer this problem by calculating the tension at each point and using function analysis etc. But there is a much easier way to find the answer. First of all, it is essential to notice that the horizontal tension in any point is constant. Just take a small part of the chain and analyse each tension to the x-axis. Since this small part is in equilibrium, the horizontal tensions are equal and constant to a certain value. That is Tcosθ = C, where θ is the angle of the tangent and the horizontal level. So T = C/cosθ. Thus, the smaller the cosθ, the bigger the tension. Or the bigger the value of θ, the bigger the tension. The angle θ is bigger at point A than at point B, and the angle at B is bigger than the angle at C. So T(A) > T(B) > T(C). Of course if we know...

Shape of a hanging rope between two points - Mathematical analysis

Let's assume that we take a rope (or a chain as shown in the picture) and hang it up between two distinct points. We want to find the shape of the rope. What equation describes it? Here is where physics comes. Well...everything is about physics! Let's dive in. So we see that the equation that describes the shape of the rope is of a hyperbolic form!

Mathematical analysis of the motion of a homogenous rotating hemisphere

Assume a homogenous hemisphere of radius R, that can rotate around a horizontal axis, inside the homogenous gravitational field of Earth ( g ) . If we define a coordinate system, starting from point O(0,0,0) as shown in the figure below, then the axis of rotation which is parallel to y'y and lies on the plane z=0 , has an abscissa x=ε (it passes through point A(ε,0,0)) • We want to find what should the distance ε from O be, in order for the initial angular acceleration of the hemisphere is the maximum possible (assume that initially the hemisphere is horizontal). • If θ is the angle of the axis x′x (as shown in figure) and the vertical, then prove that for small angles θ, the hemisphere oscillates harmonically. We want to find the value of ε, so that the period of oscillation is minimum.

A Falling Chain - Maximum reaction force

A homogenous chain of mass density ρ and length L is kept vertical above a horizontal level. We let the chain free. What is the maximum reaction force of the horizontal level ? Well, we can solve this problem easily by writing down Newton's II Law for the chain. That is F - ρLg = mυ' + m'υ. But if we name x the height of the chain x=x(t), then the mass m that goes into the momentum is m=ρx. Thus, F - ρLg = (ρx)x'' + ρυ^2. The acceleration x'' is equal to the gravitational acceleration (x''=-g) and thus the velocity υ is given by υ^2 = 2g(L - x). So plugging that into II Law, we get F = 3ρg( L - x). So the maximum force is when x = 0, and is equal to F=3ρgL=3Mg. The reaction force is maximum just before the chain stops, and immediately after the reaction force balances the weight of the chain (F=Mg).

Steven Weinberg - Nobel Prize winning physicist dies at 88

What a great physicist! I first read for him in the biography of Dimitri Nanopoulos who stated that they were very close friends and had worked many times together. Nanopoulos description of Weinberg made me curious to read more about this great theoretical physicist. Soon after I briefly read his Nobel winning paper " CONCEPTUAL FOUNDATIONS OF THE UNI- FIED THEORY OF WEAK AND ELECTROMAG- NETIC INTERACTIONS", Weinberg became on of my idols and inspirations in physics. "Professor Weinberg unlocked the mysteries of the universe for millions of people, enriching humanity's concept of nature and our relationship to the world," said Jay Hartzell, president of UT Austin, said in the statement. "From his students to science enthusiasts, from astrophysicists to public decision makers, he made an enormous difference in our understanding. In short, he changed the world." Weinberg was born in New York in 1933. His love of science began with a chemistry set, ac...