An oscillator consists of a block with mass 1.0 kg connected to a spring. It osc

Question

An oscillator consists of a block with mass 1.0 kg connected to a spring. It oscillates on
a horizontal frictionless surface according to [x is in meters]:
x(t) = 10 cos[pi/4(t-1)]
---------------------------------0------------------------->X

Determine:
(a) the amplitude, the frequency, and the period of oscillations;

(b) the spring constant;

(c) the displacement x(t), the speed v(t), the acceleration a(t), and the restoring force

F(t) at t = 4:0 s. Show on the above plot the approximate position of the block
(with respect to zero), the direction (left or right) in which it is moving, and the
direction of the restoring force;
(d) the maximum speed and at what x it occurs;

(e) the maximum acceleration and at what x it occurs;

(f) the kinetic energy, the potential energy, and the total mechanical energy at t = 4:0 s.
Bonus: What will be the angular frequency of oscillations if the mass of the block doubled
and the spring constant halved?

Show work please!

Explanation / Answer

mass m=1 kg
given eq
x(t) = 10 cos[pi/4(t-1)]
compare with x =Acos(wt-phase)
a)
amplitudde A =10 m
angular freq w=pi/4
freq f = w/2pi = 1/8 = 0.125 Hz
peroid T=1/f = 8 sec
b)
spring constant k=mw^2=1*(pi/4)^2=0.616 N/m
c)
x(t) = 10 cos[pi/4(t-1)]
velocity
v =dx/dt = -10 (pi/4)sin[pi/4(t-1)] = -7.85sin[pi/4(t-1)]
acceleration
a =dv/dt = - 10(pi/4)^2cos[pi/4(t-1)]=-6.16 cos[pi/4(t-1)]
F =ma =-6.16 cos[pi/4(t-1)]
at t=4s
F =4.35 N
it is moving in left side as velocity is negative
and restoring force F is along right side
d)
maximum speed =-7.85 m/s
e)
max acceleration a=-6.16 m/s^2
f)
velocity
v =-7.85sin[pi/4(t-1)] at 4sec
v =-5.55
Kinetic energy =0.5mv^2 =15.44 J
potential energy
position at t =4 sec
x(t) = 10 cos[pi/4(t-1)] =-7.06 m
U =0.5kx^2 =30.726 J
total energy = U+KE =30.726 +15.44 =46.16 J

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