During a tennis serve, a racket is given an angular acceleration of magnitude 155 rad/s2. At the top of the serve, the racket has an angular speed of 13 rad/s. If the distance between the top of the racket and the shoulder is 1.5 m, find the magnitude of the total acceleration of the top of the racket.

To find the magnitude of the total acceleration of the top of the racket, we need to consider both the linear and the tangential acceleration.

First, let's find the linear acceleration. The linear acceleration is given by the formula:

linear acceleration = angular acceleration × distance

Substituting the given values into the equation:

linear acceleration = 155 rad/s^2 × 1.5 m
= 232.5 m/s^2

Next, let's find the tangential acceleration. The tangential acceleration is the component of acceleration in the direction of the motion at any point along the circular path. It is given by the formula:

tangential acceleration = radius × angular acceleration

In this case, the radius is the distance between the shoulder and the top of the racket, which is 1.5 m. Thus:

tangential acceleration = 1.5 m × 155 rad/s^2
= 232.5 m/s^2

Now, to find the magnitude of the total acceleration, we can use the Pythagorean theorem, as the linear and tangential acceleration are perpendicular to each other:

total acceleration = √(linear acceleration^2 + tangential acceleration^2)
= √((232.5 m/s^2)^2 + (232.5 m/s^2)^2)
= √(54,006.25 m^2/s^4 + 54,006.25 m^2/s^4)
= √(108,012.5 m^2/s^4)
≈ 328.97 m/s^2

Therefore, the magnitude of the total acceleration of the top of the racket is approximately 328.97 m/s^2.