Please explain this to me with equations etc... A 2.50kg mass on a spring has di

Question

Please explain this to me with equations etc...

A 2.50kg mass on a spring has displacement as a function of time given by the equation
x(t)=(7.40cm)cos[(4.16rad/s)t?2.42rad].

Part A

Find the time for one complete vibration.

1.51

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Correct

Part B

Find the force constant of the spring.

43.3

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Correct

Part C

Find the maximum speed of the mass.

0.308

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Correct

Part D

Find the maximum magnitude of force on the mass.

3.20

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Correct

Part E

Find the position of the mass at t=1.00s;

?1.25

1.51

Explanation / Answer

x(t)=(7.40cm)cos[(4.16rad/s)t?2.42rad]

a) at t = 1 s

   x(t)=(7.40cm)cos[(4.16rad/s)*1?2.42rad] = 7.4cos ( 4.16 -2.42) =7.4 cos(1.74 rad) = -1.246 cm

b) V= velocity = dx/dt = - 7.4 *4.16 sin[(4.16rad/s)*1?2.42rad]   = - 30.78sin[(4.16rad/s)*1?2.42rad]

at t =1 s

V =-30.78sin[(4.16rad/s)*1?2.42rad] = -30.78 sin(1.74)= -30.34 cm/s

speed = |v| = 30.34 cm/s

c) a = acceleration = dv/dt

   a = - 30.78*4.16 cos[(4.16rad/s)*1?2.42rad] = -128cos[(4.16rad/s)*1?2.42rad]

at t =1s ,   a =-128 cos(1.74) = 21.55 cm/s^2

magniyude of acceleration = |a| = 21.55 cm/s^2

d) F = magnitude of force = ma =2.5 * 21.55 = 53.88   kg cm/s^2 = 0.53 N

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