Exercise 1. Show that the force of gravity is central.(Note that this problem cannot be done without using the definitionF = - Gmym2f of the force of gravity given on the previous page.)Let a particle of mass m move under the influence of a force Femminating from ỡ. Let:F(t) = the vector from ở to the particle at time tT(t) = F"(t) = the velocity of the particle at time ta(t) = F"(t) = the acceleration of the particle at time tp(t) = mū(t) = the momentum of the particle at time t.I(t) = F(t) x p(t) = the angular momentum.In handwritten work the notation r =F is common and convenient.

A force is called central force, when the direction of the force acting on a particle is always towards a fix point called 'center of force'. Let the vector connecting the center of force and the particle be r→ If a force is conservative, then, its cross-product with the unit vector r∧ should be zero. The curl of a vector function A→ in spherical co-ordinates is given by:- ∇→×A →= r∧r θ∧r sinθ ϕ∧∂∂r∂∂θ∂∂φArrAθrsinθAϕThe curl of the gravitational force can be given by:- ∇→×F →= r∧r θ∧r sinθ ϕ∧∂∂r∂∂θ∂∂φ-Gm1m2r200 or ∇→×F→ = 0 r∧ -rθ ∧0+Gm1m2∂1r2∂ϕ + r sinθϕ^ θ ∧0+Gm1m2∂1r2∂θ or ∇→×F→ =0 Which is the required condition for a force to be central force.

Answered: Exercise 1. Show that the force of…