University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

204

Pages:

5
Author:

Jabin F.

Degrees of Freedom (d.o.f.) d.o.f. is the number of independent variables required to describe the dynamics of a system. If a particle moves along a predefined line, then d.o.f. = 1. If a particle moves on a surface, then d.o.f. = 2. If a particle moves in space, then d.o.f. = 3. Example: If a particle moves on a sphere surface, then d.o.f. of each particle is 2, and total d.o.f. is 4. If there are many independent particles, then total d.o.f. is equal to the sum of d.o.f. of individual particles. If there are geometrical constraints/conditions among the particles, then total d.o.f. is equal to individual d.o.f. minus the number of constraints. Example: If we say that distance between two particles is constant, then there is one geometrical constraint between particles. A spring doesn't put any constraint. Example: Two particles are rigidly connected and allowed to move in space, then d.o.f. (total) = individual d.o.f. - number of constraints = 3 + 3 - 1 = 5. Rigid Bodies If many particles are connected together rigidly, then it forms a rigid body. If a rigid body moves in 3D space, then d.o.f. = 6 (3 for translational motion and 3 for rotational motion). Note: This value of d.o.f. cannot be calculated by considering d.o.f. of individual particles and constraints. If a rigid body rotates about a fixed point, then d.o.f. = 3. If a rigid body rotates about a fixed axis, then d.o.f. = 1. If a rigid body moves freely in a d-dimension space, then d.o.f. = d(d+1)/2. Example: A disk is moving in the x-direction, and its plane lies in the x-y plane. What is its d.o.f.? Solution: Linear and angular velocity both have one component. So, d.o.f. = 1 + 1 = 2. 5. Paint of Suspension moving horizontally with acceleration a0 -> Here bob has two velocities, v due to oscillation and u due to acceleration a0. Thus net velocity of bob is given by combining law. u = a0t v = lθ̇ KE = 1/2 m(u^2 + v^2 + 2uvcosθ) PE = -mglcosθ EOM: d/dt(d/dt(L)) = 0 d/dt(ml^2θ̈ + mlθ̇^2cosθ) - (-mlθ̇^2sinθ - mgl sinθ) = 0 mlθ̈ + mlθ̇^2cosθ + mlθ̇^2sinθ + mgl sinθ = 0 θ̈ + (a0/l)cosθ + (g/l)sinθ = 0 Let a0 = Asinφ, θ̇ = Acosφ such that A = √(a0^2 + g^2)/l θ̈ + A sinφcosθ + A cosφsinθ = 0 θ̈ + Asin(φ+θ) = 0 θ̈ = -(√(a0^2 + g^2)/l)sin(φ+θ) For small oscillations, sin(φ+θ) ≈ φ+θ θ̈ = -(√(a0^2 + g^2)/l)(φ+θ) This is SHM about φ with ω = √(√(a0^2 + g^2)/l) T = 2π/ω = 2π√(l/√(a0^2 + g^2))