In a fast-pitch softball game the pitcher is impressive to watch, as she delivers a pitch by rapidly whirling her arm around so that the ball in her hand moves in a circle. In one instance, the radius of the circle is 0.622 m. At one point on this circle, the ball has an angular acceleration of 68.3 rad/s2 and an angular speed of 15.8 rad/s. (a) Find the magnitude of the total acceleration (centripetal plus tangential) of the ball. (b) Determine the angle of the total acceleration relative to the radial direction.
I know what (a) is but can't figure out (b). (a) is 160.27m/s^2
To find the angle of the total acceleration relative to the radial direction, we need to understand the components of the total acceleration: centripetal acceleration and tangential acceleration.
Centripetal acceleration (ac) is the acceleration directed towards the center of the circular path and can be calculated using the formula:
ac = ω²r
where ω is the angular speed and r is the radius of the circle. In this case, the angular speed is given as 15.8 rad/s and the radius is 0.622 m. Therefore, we can find the centripetal acceleration (ac):
ac = (15.8 rad/s)² * 0.622 m
ac = 157.7124 m/s²
Tangential acceleration (at) is the acceleration along the tangent to the circular path. In this case, the angular acceleration (α) is given as 68.3 rad/s². The tangential acceleration can be calculated using the formula:
at = αr
where α is the angular acceleration and r is the radius of the circle. Plugging in the given values:
at = (68.3 rad/s²) * 0.622 m
at = 42.4406 m/s²
Now, to find the total acceleration (a), we need to find the vector sum of the centripetal acceleration and the tangential acceleration. We'll use the Pythagorean theorem to find the magnitude of the total acceleration:
a = √(ac² + at²)
a = √((157.7124 m/s²)² + (42.4406 m/s²)²)
a = 160.27 m/s²
At this point, you correctly found the magnitude of the total acceleration (a) as 160.27 m/s².
To determine the angle of the total acceleration relative to the radial direction, we can use trigonometry. The angle (θ) can be calculated using the inverse tangent (arctan) function:
θ = arctan(at/ac)
Plugging in the known values:
θ = arctan(42.4406 m/s² / 157.7124 m/s²)
θ ≈ 15.436 degrees
Therefore, the angle of the total acceleration relative to the radial direction is approximately 15.436 degrees.