Question

The position of a particle attached to a vertical spring is given by y = (y0 cos ωt)j. The y axis points upward, y0 = 32.1 cm, and ω = 21.44 rad/s. (a) Find the displacement of the particle during the time interval from t = 0 to t = 8.5 s. (Round your answer to the nearest integer.)

(a) Find the displacement of the particle during the time interval from t = 0 to t = 8.5 s. (Round your answer to the nearest integer.)

(b) Find the distance (in centimeters) the particle traveled during this time interval.

(c) Many physical systems are modeled by a particle attached to a spring. List some examples of systems that may be modeled by springs. It may be helpful to use the index of this book or the Internet.

Answer 1

The position of a particle attached to a vertical spring is given by -

(t) = (y₀ cos t)

where, y₀ = initial position of a particle = 32.1 cm

= angular speed of a particle = 21.44 rad/s

then, we get

(t) = [(32.1 cm) cos (21.44 rad/s) t]

At t = 0 s, the distance traveled by a particle which will be given as -

(0) = [(32.1 cm) cos (21.44 rad/s) (0 s)]

(0) = [(32.1 cm) cos (0 rad)]

(0) = (32.1 cm) (1 rad)]

(0) = (32.1 cm)

At t = 8.5 s, the distance traveled by a particle which will be given as -

(8.5) = [(32.1 cm) cos (21.44 rad/s) (8.5 s)]

(0) = [(32.1 cm) cos (182.24 rad)]

(8.5) = (32.1 cm) (0.9996 rad)]

(8.5) = (32.08 cm)

Therefore, displacement of a particle during the time interval from t=0 to t=8.5 sec which will be given by -

= (8.5) - (0)

= [(32.08 cm) - (32.1 cm)]

= - (0.02 cm)