Question

A particle rotates counterclockwise in a circle of radius 3.00 m
with a constant angular speed of 8.00 rad/s. At t = 0, the
particle has an x coordinate of 2.00 m and is moving to the
right. A)Determine the x coordinate as a function of time.

please I want to know why we use phi = - 0.841 in minus not plus?

Answer 1

• -0.841 rad.The X coordinate as a function of time .

Answer 2

Given: The$radius=3.00m.$, an angular speed of  $8.00$rad/s. At t$=0$

We have to find the$x$ coordinate as a function of time.

As we know that the amplitude of the particle’s motion equals the radius of the circle and$\omega=8.00 \mathrm{rad} / \mathrm{s}$.

We have,

$x=A \cos (\omega t+\phi)=(3.00 \mathrm{~m}) \cos (8.00 t+\phi)$

We can evaluate ϕ  by using the initial condition that is $x=2.00m. at( t=0)$

Now,

$2.00 \mathrm{~m}=(3.00 \mathrm{~m}) \cos (0+\phi) \phi=\cos ^{-1}\left(\frac{2.00 \mathrm{~m}}{9.00 \mathrm{~m}}\right)$

$Here \phi=48.2^{\circ} \text {, then the coordinate } x=(300 \mathrm{~m}) \cos \left(8.00 t+48.2^{\circ}\right)$

$\text { would be decreasing at time } \mathrm{t}=0$

$\text { Because our particle is first }\text { moving to the right, we must choose } \phi=-48.2$

Hence, $\text { The } x \text { coordinate as a function of time will be }$

$x=(3.00 \mathrm{~m}) \cos (8.00 t-0.841)$

$\text { here } \phi \text { in the cosine function therefore it must be in radians i.e.\text { -0.841 rad. } }$