Physics Department, Shiv Nadar University (PHY101-2015) Tutorial-8

1. The potential energy of a particle is 𝑈(𝑥) = −𝑥3 + 2𝑥2 + 3𝑥. Find equilibrium

points and determine if they are stable or unstable. Also show the equilibrium

points by plotting U(x).

2. A particle of mass m is constrained to move along the positive x-axis under the

influence of a single force whose potential energy function is

U(x)=−a x2exp(-x

2/b

2)

where a, b are positive constants. (a) Find the equilibrium point(s). (b) For each

stable equilibrium, calculate the frequency of small oscillations.

3. A block of mass M slides down a plane of angle θ. Find the speed of the block

after it has descended through height h, assuming that it starts from rest. The

coefficient of friction μ varies as 2μ0 s (μ0 is a constant), where s is measured along

the incline, with s=0 taken at the initial position of the block.

4. A rod of length L has a non-uniform density. The mass per unit length of the rod,

λ, varies as λ = λ0(x/L), where λ0 is a constant and x is the distance from the end

marked 0. Find the center of mass.

5. Uniform Triangular Plate. Consider the two-dimensional case of a uniform right

triangular plate of mass M, base b, height h, and small thickness t.