Question

A coin with a diameter of 2.06 cm is dropped on edge onto a horizontal surface. The coin starts out with an initial angular speed of 15 rad/s and rolls in a straight line without slipping. If the rotation slows with an angular deceleration of magnitude 1 rad/s^2. How far does the coin roll before coming to rest?

Answer 1

The coin rolls a distance of 3.24 cm before coming to rest.

Step-by-step explanation:

To find the distance the coin rolls before coming to rest, we need to determine the time it takes for the coin to stop rotating.

We can use the equation θ = ωit + (½)αt2 to find the angle by which the coin rotates before coming to rest.

Since the coin starts from rest and has an angular deceleration of magnitude 1 rad/s^2, we can rearrange the equation to solve for t: t = sqrt(2θ/α).

Substituting the given values, we have t = sqrt(2 * 15 rad / 1 rad/s^2) = sqrt(30) seconds.

To find the distance rolled by the coin, we use the equation s = rθ, where s is the distance, r is the radius of the coin, and θ is the angle.

Substituting the values, we have s = (2.06 cm/2) * 2 * π = 3.24 cm.

So, the coin rolls a distance of 3.24 cm before coming to rest.

Answer 2

To solve this problem we will apply the kinematic equations of angular motion for which the square of the angular velocity change is described is equivalent to twice the angular acceleration by time. This is,

`\omega_2^2 - \omega_1^2 = 2\alpha \theta`

Here,

`\omega_(1,2)`= Final and initial Angular Velocity

`\alpha`= Angular acceleration

`\theta`= Angular displacement

Our values are given as,

`D = 2.06cm=0.0206m \rightarrow r = (d)/(2) = 0.0103m`

`r = 0.0103 m`

`\omega_1 = 15rad/s`

`\omega_2 = 0rad/s \rightarrow` When is coming to rest

`\alpha = -1 rad/s^2`

Replacing we have that,

`\omega_2^2 - \omega_1^2 = 2\alpha \theta`

`\theta = ((\omega_2^22 - \omega_1^2 ))/(2\alpha)`

`\theta= ((0 - (15 rad/s)^2 ))/(2(-1 rad/s^2))`

`\theta= 112.5 rad`

Through the arc length ratio the distance in meters is

`x = r\theta`

`x = (0.0103 m)(112.5 rad)`

`x = 1.158 m`

Therefore the coin will roll 1.158m before coming to rest.