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

2024
Views:

98

Pages:

5
Author:

Jabin F.

# Potential Energy It is a scalar representation of conservative force. Mathematically ∇ x F = 0 for conservative force as ∇ x (scalar function) = 0 always ∴ F = -∇U must be. ∴ F = -∇U → F = ±∇U # In Physics where -ve sign is taken conventionally. # -ve sign means direction of force is from higher potential energy region to lower potential energy region. General formula of gradient of a scalar function 'u' ∇u = (∂u/∂x)î + (∂u/∂y)ĵ + (∂u/∂z)k̂ in cartesian co-ordinate → ∇u = (∂u/∂x)î + (∂u/∂y)ĵ + (∂u/∂z)k̂ in spherical polar co-ordinate → ∇u = (∂u/∂r)ȓ + (1/r)(∂u/∂θ)θ̂ + (1/rsinθ)(∂u/∂φ)φ̂ Example: Potential energy of a system is written in spherical polar co-ordinate as U = k cosθ. Calculate force. Solution: U = k cosθ F = -∇U = -[(∂U/∂r)ȓ + (1/r)(∂U/∂θ)θ̂ + (1/rsinθ)(∂U/∂φ)φ̂] = -[k cosθȓ - k sinθθ̂] F = k[sinθȓ - cosθθ̂] Q: A conservative force is given in spherical polar co-ordinates as F = (A/r²) (α cosθ ȓ̂ + β sinθ θ̂) Find the value of α. Solution: ∇ x F = (1/r²sinθ) | ∂/∂r ∂/∂θ ∂/∂φ | | r²sinθ r sinθ 0 | | Aαcosθ Aβsinθ 0 | must be zero. ∴ (1/r²sinθ) x r sinθ [βA sinθ - αA sinθ] φ̂ = 0 Which is only possible for: [β + α] = 0 α = -2