An object, attached to the end of a horizontal spring, oscillates with simple ha

Question

An object, attached to the end of a horizontal spring, oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = (5cm) sin (2t ). What are: a) The position of the object at t=1.0 s b) The velocity (in m/s) of the body at t = 1.0 s c) The acceleration of the body at t=1.0 s. d) Amplitude of the oscillation. e) Period of the oscillation. f) Maximum speed of the body. An object, attached to the end of a horizontal spring, oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = (5cm) sin (2t ). What are: a) The position of the object at t=1.0 s b) The velocity (in m/s) of the body at t = 1.0 s c) The acceleration of the body at t=1.0 s. d) Amplitude of the oscillation. e) Period of the oscillation. f) Maximum speed of the body. An object, attached to the end of a horizontal spring, oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = (5cm) sin (2t ). What are: a) The position of the object at t=1.0 s b) The velocity (in m/s) of the body at t = 1.0 s c) The acceleration of the body at t=1.0 s. d) Amplitude of the oscillation. e) Period of the oscillation. f) Maximum speed of the body.

Explanation / Answer

x= 5(cm) sin(2t)
Converting it into m
x= 0.05(m) sin(2t)
or
x= 0.05 sin(2t)

a)
at t= 1.0 s
x= 0.05 sin(2*1)
   = 0.05 sin(2)   here 2 is in radians which corresponds to 114.6 degree
   = 0.05 sin (114.6)
   =0.046 m
   =4.6 cm

b)
x= 0.05 sin(2t)
velocity = dx/dt = 0.05*2 cos (2t)
v = 0.1*cos (2t)
at t=1.0 s
v = 0.1*cos (2*1)
   = 0.1* cos(2)   here 2 is in radians which corresponds to 114.6 degree
   = 0.1 cos (114.6)
   = -0.042 m/s

c)
v = 0.1*cos (2t)
a = dv/dt = - 0.1 *2 * sin(2t)
   = -0.2 sin(2t)
at t = 1.0 s
a= -0.2 sin(2*1)
   = -0.2 sin(2)   here 2 is in radians which corresponds to 114.6 degree
   = -0.2 sin (114.6)
   = -0.182 m/s^2

d)
Since x= 0.05 sin(2t)
Comparing with general form x= A*sin(w*t)
Amplitude = 0.05 m = 5 cm

e)
Since x= 0.05 sin(2t)
Comparing with general form x= A*sin(w*t)
w=2
2*pi/T = 2
T = pi seconds
Period = pi seconds = 3.14 seconds

f)
Since, v = 0.1*cos (2t)
Maximum speed will be 0.1 m/s

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