Equilibrium

Static equilibrium

Problem:

Two telephone poles are separated by 40 m and connected by a massless wire.  A bird of mass 0.5 kg lands on the wire midway between the poles, causing the wire to sag 2.0 m below horizontal.  What is the tension in the wire?

Solution:

Problem:

One end of a spring of negligible mass is attached to the ceiling.  When a 250 g mass is placed on the free end (without stretching the spring) and then released, the mass descends 20.0 cm before it changes direction and begins to ascend.  What is the spring constant k (in N/m)?

Solution:

Problem:

Two identical springs with spring constant k = 1 N/m and equilibrium length l = 0.25 m are connected by a middle string of length L = (3/8) m and support a weight w = 0.5 N. 
Two strings of length 1 m are loosely connected as shown. 
Find the position of the weight below the support and show that it moves up when the middle string is cut.

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Solution:

Problem:

Consider a particle moving along the x axis under the influence of the potential

U(x) = ½k(x/a)2 - ⅓k(x/a)3

where k and a are constants. 
(a) Plot the potential. 
(b) Find the equilibrium point(s).  
(c) Determine whether the equilibrium point(s) are stable or unstable.

Solution:

Problem:

Two small positively charged spheres are suspended from a common point at the ceiling by the insulating light strings of equal length.  The first sphere has mass m1 and charge q1 while the second one has mass m2 and charge q2.  If the first string makes an angle θ1 with the vertical, find the angle θ2 that the second string makes with the vertical.

Solution:

Dynamic equilibrium

Problem:

A hollow sphere of radius R = 0.5 m rotates about a vertical axis through its center with an angular velocity of ω = 5/s.   Inside the sphere a small block is moving together with the sphere at the height of R/2.   (Let g = 10 m/s2.)
(a)  What is the minimal coefficient of friction to fulfill this condition?
(b)  Find the minimal coefficient of friction for the case of ω  = 8/s.
(c)  Given the minimal coefficient of friction, investigate the problem of stability in both cases,
(i)  for a small change of the position of the block,
(ii)  for a small change of the angular velocity of the sphere.
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Solution: