CD Player question help

Question

While placing a compact disc into a CD player, you notice a small chip on its edge. You attempt to play the CD anyway by placing the CD into the player's deck with the chip at 𝜃0=16.5∘�0=16.5∘ as measured counterclockwise from the +𝑥+�‑axis. The CD begins to rotate counterclockwise with angular acceleration 𝛼=2.31 rad/s2.�=2.31⁢ rad/s2.

If the CD has been spinning for 𝑡=3.51 s�=3.51 s and the disc has a radius of 𝑟=6.00 cm,�=6.00 cm, what are the 𝑥� and 𝑦� coordinates of the chip after this time, assuming the center of the disc is located at (0.00,0.00).

Answer 1

This question involves three issues:

1) θ₀ is given in degrees, while the angular acceleration is given using radians.

We just need to be consistent, and I will chose to use radians.

The conversion between degrees and radians is given by

360° = 2π rad

2) You need the kinematic equation

θ = θ₀ + ω₀t + αt²/2 (1)

If you had calculus, you can derive it yourself, otherwise you need to remember it.

3) The relationship between an angle θ and x, y- coordinates is

x = rcos(θ) (2)

y = rsin(θ) (3)

Here is how you apply them to your situation.

Let

θ₀ = 16.5° = 16.5×2π/360 rad be the initial angle,

ω₀ = 0 be initial angular velocity

α = 2.31 rad/s² be the constant angular acceleration,

t = 3.51 s be the time the CD has been spinning,

r = 6.00 cm be the radius,

θ (unknown) be the angle after time t.

Plug the given quantities into (1) to get

θ = 16.5×2π/360 + 0×3.51 + 2.31×3.51²/2 ≈ 14.5

Then use (2) and (3) to get

x = 6×cos(14.5) = -2.13 cm

y = 6×sin(14.5) = 5.6 cm