Question

Consider a block attached to a spring in SHM with ?o=+?/4 and T=8 with A =10m.

a) What is x(0), x(1), x(2 seconds)...? What is the equation for x(t)?

c) what is the velocity at t=0? 1 sec?...

Answer 1

Given a block attached to a spring in SHM.

To verify expression for Displacement and Velocity when calculated directly from SHM -

• Displacement in SHM is = A cos (?t + ?)
• Velocity v(t) in SHM is = -Aw sin (?t + ?)

• Using x(t)= A cos (?t + ?)

• Acceleration ,a(t) = dv/dt

• a(t) = - ?²A cos (?t + ?)
• a(t) = - ?²x(t)         Using x(t)= A cos (?t + ?))
• Here w = 2

Calculation of displacement at x(0), x(1), x(2)

x(0) = A cos (2t+/4)

       = 10 cos (2*0+/4)

       = 10 cos /4

       = 10 * 1/ = 7.072 m

x(1) = A cos (2t+/4)

       = 10 Cos (2 + /4)

       = 10 Cos /4

       = 10 * 1/

      = 7.072 m

x(2) = A cos (2t+/4)

       = 10 Cos (2*2 + /4)

       = 10 cos (4 + /4)

      = 10 cos /4

      = 7.072 m

Calculation of Velocity v(o), v(1)

Velocity v(t) = -Aw sin (?t + ?)

              v(0) = - 10 * 2 * sin (2*0 + /4)

   = -20 sin /4

   = -20*3.14 * 1/

   = - 44.413 m/s

           v(1) = - 10 * 2 * sin (2*1 + /4)

                  = -20 sin /4

                 = - 44.413 m/s