What is the tangential acceleration of a bug on the rim of a 12.0 in. diameter disk if the disk moves from rest to an angular speed of 75 rev/min in 4.0 s?
First calculate the angular acceleration, alpha , in radians/s^2.
w(final) = 75 rpm *2 pi rad/rev/60 s/min = 7.85 rad/s
alpha = w(final)/4.0 s = ___ rad/s^2
Tangential acceleration = R * alpha = __ m/s^2
I suggest using metric units (R = 0.1524 m), to get the tangential acceleration in m/s^2. If you use R = 6 inches, a will be in in/s^2.
To find the tangential acceleration of the bug on the rim of the disk, we need to use the formula for tangential acceleration:
Tangential acceleration (at) = Radius (r) * Angular acceleration (α)
The radius of the disk is half of its diameter, so the radius (r) is 12.0 in / 2 = 6.0 in.
To find the angular acceleration (α), we first need to convert the given angular speed from rev/min to rad/s.
Step 1: Convert angular speed from rev/min to rad/s
Given: Angular speed (ω) = 75 rev/min
Conversion factor: 1 rev = 2π rad
Conversion factor: 1 min = 60 s
Angular speed (ω) in rad/s = 75 rev/min * (2π rad/1 rev) * (1 min/60 s) = 75 * (2π/60) rad/s = 5π/4 rad/s
Now we have the angular speed (ω) in rad/s.
Step 2: Calculate the angular acceleration (α)
We are given the time (t) = 4.0 s, and we can use the following formula to calculate the angular acceleration:
Angular acceleration (α) = Change in angular speed (Δω) / Change in time (Δt)
In this case, the object starts from rest, so the initial angular speed (ω₀) is 0.
Change in angular speed (Δω) = Final angular speed (ω) - Initial angular speed (ω₀) = ω - ω₀ = ω
Angular acceleration (α) = ω / Δt = (5π/4 rad/s) / 4.0 s = (5π/4) / 4 rad/s² = (5π/16) rad/s²
Now we have the angular acceleration (α) in rad/s².
Step 3: Calculate the tangential acceleration (at)
Using the formula at = r * α, we can calculate the tangential acceleration.
Tangential acceleration (at) = 6.0 in * (5π/16) rad/s² = (30π/16) in/s² ≈ 5.89 in/s² (rounded to two decimal places)
Therefore, the tangential acceleration of the bug on the rim of the 12.0 in. diameter disk is approximately 5.89 in/s².