13. The constant angular acceleration of a pulley is 4 rad/s2 and its angular speed its 3 rad/s. Determine the radius of the pulley if the total acceleration of a print on the rim of the pulley is 3.5m/s2.

I assume that you mean a point on the rim, not a print. The radial and tangential acceleration components are perpendicular. Therefore

(3.5)^2 = (4 R)^2 + [ 3^2*R]^2
The first term on the right is the square of the tangential acceleration. The second term is the square of the centripetal acceleration.
Solve for R.

13. The constant angular acceleration of a pulley is 4 rad/s2 and its angular speed its 3 rad/s. Determine the radius of the pulley if the total acceleration of a print on the rim of the pulley is 3.5m/s2.

To determine the radius of the pulley, we can use the following equations:

1. Angular acceleration (α) = Change in angular speed (Δω) / Time taken (Δt)
2. Δω = ωf - ωi
3. Tangential acceleration (at) = Radius (r) * Angular acceleration (α)
4. Total acceleration (atotal) = √(at^2 + ar^2)

Given:
- Angular acceleration (α) = 4 rad/s^2
- Angular speed (ω) = 3 rad/s
- Total acceleration (atotal) = 3.5 m/s^2

We need to convert the angular speed (ω) from rad/s to rad/s^2 since the equation requires the rate of change of angular speed.

Step 1: Calculate Δω
Δω = ωf - ωi
Δω = 0 - 3 rad/s
Δω = -3 rad/s

Step 2: Calculate Δt
Δt = Δω / α
Δt = (-3 rad/s) / (4 rad/s^2)
Δt ≈ -0.75 s (negative sign indicates a decrease in angular speed)

Step 3: Calculate at
at = r * α
3.5 m/s^2 = r * 4 rad/s^2
r = 3.5 m/s^2 / 4 rad/s^2
r ≈ 0.875 m

Therefore, the radius of the pulley is approximately 0.875 meters.

To determine the radius of the pulley, we need to use the following equations:

1. Angular acceleration (α) is the rate at which the angular velocity (ω) of an object changes over time.

α = (change in ω) / (change in time)

2. Linear acceleration (a) is the acceleration of an object moving in a straight line.

a = α * r

3. The total acceleration of a point on the rim of a pulley is the sum of its linear acceleration (tangential acceleration) and its centripetal acceleration.

a_total = a_tangential + a_centripetal

Let's solve the problem step by step:

Given:
Constant angular acceleration (α) = 4 rad/s^2
Angular speed (ω) = 3 rad/s
Total acceleration (a_total) = 3.5 m/s^2
We need to find the radius (r) of the pulley.

First, let's find the tangential acceleration (a_tangential):

a_tangential = α * r
3.5 m/s^2 = (4 rad/s^2) * r

Next, let's find the centripetal acceleration (a_centripetal):

a_centripetal = ω^2 * r
a_centripetal = (3 rad/s)^2 * r

Since the pulley has a constant angular acceleration, we can use the following equation relating the angular acceleration, angular speed, and time:

ω = ω_0 + α * t

Given that the angular speed (ω_0) is 0 rad/s (initial speed) and the angular speed (ω) is 3 rad/s, we can rearrange the equation to find the time (t):

3 rad/s = 0 rad/s + (4 rad/s^2) * t
t = 3 rad/s / 4 rad/s^2
t = 0.75 seconds

Now, we can solve for the radius (r) using the equations for tangential acceleration and centripetal acceleration:

a_tangential = (4 rad/s^2) * r
a_centripetal = (3 rad/s)^2 * r

Since a_total = a_tangential + a_centripetal:

3.5 m/s^2 = (4 rad/s^2) * r + (3 rad/s)^2 * r

Now, we can solve this equation to find the value of r, the radius of the pulley.