A particle moves so that position x in meters is given as a function of time t i

Question

A particle moves so that position x in meters is given as a function of time t in seconds by the equation x(t) = c1 cos(c2t + c3), where c1 = .0296 m, c2 = 323 s ^-1 , c3 = 1.05 . Give numerical values for the following:

(a) Amplitude (m)

(b) Angular frequency (rad/s)

(c) Frequency (Hz)

(d) Period (s)

(e) Phase constant (rad) (f) What is the position (m) of the particle at t = 3.00 ms?

(g) What is the velocity (m/s) of the particle at t = 3.00 ms?

(h) What is the acceleration (m/s2) of the particle at t = 3.00 ms?

Explanation / Answer

here this equation is same as

X(t) = X(m) cos(wt +Q)

by the comparision

a. amplitude is

X(m) = C1 = 0.0296 m Ans

b. angular frequency

W = C2 = 323 rad/s Ans

c. frequency is

F =W / 2 x 3.14

= 323 / 2 x 3.14

= 51.43 Hz Ans

d. period is

T = 2 x 3.14 / W

= 2 x 3.14 / 323

=0.01944 s Ans

e. phase constant

Q = C2 = 1.05

and at t = 3ms

X = 0.0296 cos( 323 x 3 x10-3 + 1.05)

= -0.01282 m here negative sign represent only direction of motion

g. as we know that

V = dX / dt

= - 0.0296 x 323 x sin2.019

= - 8.61645 m/s Ans

h. we know that

a = dV / dt

= c1 c2 c2 cos( c2t + c3)

= -1338.236 m/s2 Ans

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