To find the total linear acceleration of a point on the outer edge of the wheel at $t = 0.60 \, \text{s}$, we need to calculate two components of acceleration:

1. Tangential acceleration ($a_t$): This is due to the angular acceleration of the wheel.
2. Centripetal acceleration ($a_c$): This arises because the point on the wheel is moving in a circular path.

The total linear acceleration is the vector sum of these two components:

$a_{\text{total}} = \sqrt{a_t^2 + a_c^2}$

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Step 1: Calculate Tangential Acceleration ($a_t$)

The tangential acceleration is given by:

$a_t = \alpha r$

where:

• $\alpha = 3.0 \, \text{rad/s}^2$ (angular acceleration),
• $r = \frac{\text{diameter}}{2} = \frac{40 \, \text{cm}}{2} = 0.20 \, \text{m}$ (radius).

$a_t = 3.0 \times 0.20 = 0.60 \, \text{m/s}^2$

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Step 2: Calculate Centripetal Acceleration ($a_c$)

The centripetal acceleration is given by:

$a_c = \omega^2 r$

where:

• $\omega$ (angular velocity) at $t = 0.60 \, \text{s}$ is found using:

$\omega = \alpha t$

Substituting $\alpha = 3.0 \, \text{rad/s}^2$ and $t = 0.60 \, \text{s}$:

$\omega = 3.0 \times 0.60 = 1.8 \, \text{rad/s}$

Now calculate $a_c$:

$a_c = (1.8)^2 \times 0.20 = 3.24 \times 0.20 = 0.648 \, \text{m/s}^2$

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Step 3: Calculate Total Linear Acceleration ($a_{\text{total}}$)

Combine the components $a_t$ and $a_c$:

$a_{\text{total}} = \sqrt{a_t^2 + a_c^2}$

Substitute $a_t = 0.60 \, \text{m/s}^2$ and $a_c = 0.648 \, \text{m/s}^2$:

$a_{\text{total}} = \sqrt{(0.60)^2 + (0.648)^2} = \sqrt{0.36 + 0.419904} = \sqrt{0.779904}$

$a_{\text{total}} \approx 0.884 \, \text{m/s}^2$

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Final Answer:

The magnitude of the total linear acceleration is approximately:

$\boxed{0.88 \, \text{m/s}^2}$

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Do you want further details or have any questions? Here are 5 related questions to consider:

1. How does angular acceleration affect the tangential acceleration of a point on a rotating wheel?
2. What role does the radius play in centripetal acceleration?
3. How would the total acceleration change if the wheel’s angular acceleration doubled?
4. Why is the centripetal acceleration zero at $t = 0$?
5. What is the relationship between linear acceleration and angular motion?

Tip: Always check the units of angular velocity and acceleration to ensure proper calculations!