A car engine accelerates at a constant rate from 170 rpm to 3050 rpm in 7 seconds.

I have found that at t=0, the angular velocity (omega)=17.8 rad/s, the angular acceleration (alpha)= 43.1 rad/s^2, the tangential acceleration = 7.76 m/s^2, and the centripetal acceleration is 57 m/s^2

What angle does the net acceleration vector make with the radius?

If your tangential acceleration is 7.76

and the radial acceleration is -57 (inward),
then
tangent of angle between acceleration vector and vector pointing toward circle center = 7.76/57
so that angle = 7.75 degrees
since the radius vector is directed outward we need 180 - 7.75

To find the angle that the net acceleration vector makes with the radius, we can use the concept of centripetal acceleration and tangential acceleration.

The net acceleration vector can be visualized as the vector sum of the centripetal acceleration and the tangential acceleration.

Since the centripetal acceleration is directed towards the center of the circular path, it is perpendicular to the radius of the circle. Therefore, the centripetal acceleration vector makes a right angle (90 degrees) with the radius.

On the other hand, the tangential acceleration vector is in the direction of the change in velocity of the object. It is tangential to the circular path.

Now, we can determine the angle between the net acceleration vector and the radius by considering the vector addition of the centripetal acceleration and the tangential acceleration.

Using the given values:

Centripetal acceleration = 57 m/s^2
Tangential acceleration = 7.76 m/s^2

We can calculate the magnitude of the net acceleration vector:

Net acceleration = √(centripetal acceleration)^2 + (tangential acceleration)^2

Net acceleration = √(57^2 + 7.76^2) ≈ 57.67 m/s²

Now, let's denote the angle between the net acceleration vector and the radius as θ.

Using trigonometry, we can express this angle by finding the inverse tangent of the centripetal acceleration divided by the tangential acceleration:

θ = arctan(centripetal acceleration / tangential acceleration)

θ = arctan(57 / 7.76) ≈ 82.0 degrees

Therefore, the angle that the net acceleration vector makes with the radius is approximately 82.0 degrees.