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

2024
Views:

263

Pages:

5
Author:

Jabin F.

# Redundant Constraints: A constraint which is not required or not useful is called a redundant constraint. If a body moves freely in 3D space, then Redundant constraint = C - 6 (where C is the number of constraints). For a rigid body moving freely: 6 = 3N - number of constraints Number of constraints = 3N - 6 From here, we can calculate the number of constraints and put it in the formula of Redundant constraint. Let us make a rigid body with 5 particles. From the figure, we find 10 constraints and then: d.o.f. = 3N - 10 = 3 * 5 - 10 = 5, which is incorrect as d.o.f. = 6 for a rigid body. Therefore, there must be 1 redundant constraint. Example: 10 particles are connected rigidly to form a rigid body. Calculate the number of constraints required and the number of redundant constraints. Solution: d.o.f. = 3N - number of constraints Number of constraints = 3N - 6 = 3 * 10 - 6 = 24 NC (number of constraints) = 10C2 - 24 = 45 - 24 = 21 Number of redundant constraints = NC - number of constraints = 21 - 24 = -3 P-82: Hamiltonian of a simple pendulum consisting of a mass 'm' attached to a massless string of length l is H = (1/2)m(lθ̇)² + mgl(1 - cosθ). If L denotes the Lagrangian, the value of dL/dt is: Solution: H = (1/2)m(lθ̇)² + mgl(1 - cosθ) → L = (1/2)m(lθ̇)² - mgl(1 - cosθ) dL/dt = m(lθ̇)(lθ̈) - mgl(sinθ)θ̇ Now we have to find the value of θ̇ and θ̈, so we use Hamilton's equations of motion: θ̇ = ∂H/∂pθ = pθ/(ml²) θ̈ = -∂H/∂θ = -mgl(sinθ)/ml² = -g(sinθ)/l θ̈ = pθ/ml² - (mgl sinθ)/ml² = -g sinθ/l Using these values in equation (1), we get: dL/dt = ml²θ̇² - (mgl sinθ)θ̇ + mgl sinθθ̇ = (ml²θ̇² - gθ̇sinθ + gθ̇sinθ)/l = ml²θ̇²/l