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

2024
Views:

226

Pages:

5
Author:

Jabin F.

Alternate Method for finding velocity: Conservation of Energy: (KE + PE)initial = (KE + PE)instantaneous (0 + (-GmM)/R) = (1/2)mv² + (-GmM)/(R - x) (1/2)mv² = GmM(1/R - 1/(R - x)) v² = (2GM/R) * (x / (R - x)) Q: At t = 0, a constant horizontal force F is applied on a block towards right. Calculate speed as a function of displacement (x) and also calculate maximum elongation produced. Solution: Firstly, we have to calculate velocity at any instant of time which is calculated by either EOM or conservation of energy. Method 1: EOM: Fnet = ma = -kx + F ma = (-kx + F)dx m∫a dv = ∫(-kx + F)dx (1/2)mv² = Fx - (1/2)kx² + C v² = (2/m)(Fx - (1/2)kx²) + C Method 2: Conservation of Energy: Firstly, we have to calculate potential energy due to the constant force. U = -∫F dx + C = -∫(F₀) dx + C = -Fx + C Now apply conservation of energy: (KE + PE)initial = (KE + PE)final (1/2)mv₀² + 0 + C = (1/2)mv² + (1/2)kx² - Fx + C (1/2)mv₀² + C = (1/2)mv² + (1/2)kx² - Fx + C (1/2)mv₀² = (1/2)mv² + (1/2)kx² - Fx Elongation will be maximum when block comes to rest and v = 0: 0 = (1/2)kx² - Fx 2Fx - kx² = 0 x(2F - kx) = 0 Hence max. elongation: x = 0 (initial position) or x = 2F/k (final position) Note: When only a constant force is acting on a spring, then max. elongation is given as x = F/k. Example: [Diagram of a spring-mass system with a constant force] max. elongation = F/k Example: [Diagram of a spring-mass system with a variable force] max. elongation = QE/k