A ball of mass (m) is attached at the right end of a spring lying horizontally on a fixed frictionless suport.
- The ball is also subject to the downward gravitational force, where g=9.8m/s^2.
- In reaction to the downwards force the normal force (Rg) pushes back with equal force in the upwards direction.
- The left end of the spring is attached to a wall at a fixed point (A).
- The natural length of the spring is L0 and the spring constant is k>0,
- We denote the restoring force of the spring acting on the ball by (T).
- Initially the ball is not moving and the spring is unstretched.
- An external force (F(t)) is then applied to the ball, pulling it to the right.
- The damping force (R), is proportional the the instantaneous velocity of the ball, and the damping constant is denoted as B (beta symbol)
- The direction to the right is the positive direction
- The origin is taken to be the fixed point (A)
- The ball is moving along the horizontal axis ONLY
- x(t) is the position of the ball at any time (t) after the external force F(t) is applied
Part A) show all the forces acting on the ball
Part B) Apply newtons second law of motion on the horizontal axis to find the differential equation x(t) satisfes.
Part C) If this experiment were undertaken on the moon, where g=1.6m/s^2, how would this affect the motion of the ball. What would change?
Part D) answered.
Part E) Let w02=k/m and =1meter. Assume that there is no damping and that external force F(t)=F0*m*cos(w0*t). Then the equation of motion derived in Part B) reduces to:
x"(t) +w02*x(t) = F0*cos(w0*t) + w02 ...(1)i) Find the general solution of equation (1). Give a physical interpretation of your solution.
ii) Solve equation (1) assuming the initial conditions from Part D.
Part F) Suppose now the characteristics of the spring mass system are such that the equation of motion in Part B reduces to
x"(t) = 4x'(t) + 3x(t) = 2e2t*cos(t) +3te-4t+ 3
Find the general solution to this differential equation.
Absolutely any help anyone has to any of these parts would be much appreciated. I have answers to all parts and am happy to discuss but I feel they may be quite wrong