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 0.608 m. At one point on this circle, the ball has an angular acceleration of 65.3 rad/s2 and an angular speed of 19.7 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.

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Oh, it sounds like this pitcher is spinning the ball like a pro! Let's calculate the magnitude of the total acceleration (centripetal plus tangential) of the ball first.

To do this, we need to know that the total acceleration of an object moving in a circle is given by the equation:

a_total = √(a_centripetal^2 + a_tangential^2)

Where a_centripetal represents the centripetal acceleration and a_tangential represents the tangential acceleration.

The centripetal acceleration can be calculated using the formula:

a_centripetal = r * ω^2

And the tangential acceleration can be calculated using the formula:

a_tangential = r * α

Given that the radius (r) is 0.608 m, the angular speed (ω) is 19.7 rad/s, and the angular acceleration (α) is 65.3 rad/s^2, we can substitute these values into the formulas to find the magnitudes.

a_centripetal = (0.608 m) * (19.7 rad/s)^2
a_tangential = (0.608 m) * (65.3 rad/s^2)

Now, let's plug in the values and do the math.

a_centripetal = (0.608 m) * (19.7 rad/s)^2
= 235.41568 m/s^2

a_tangential = (0.608 m) * (65.3 rad/s^2)
= 39.7584 m/s^2

Now, let's calculate the magnitude of the total acceleration.

a_total = √(235.41568 m/s^2)^2 + (39.7584 m/s^2)^2
= √(55392.74180544 m^2/s^4 + 1580.43717136 m^2/s^4)
≈ √(56973.1799768 m^2/s^4)
≈ 238.745 m/s^2

So, the magnitude of the total acceleration of the ball is approximately 238.745 m/s^2.

Now, for the angle of the total acceleration relative to the radial direction, you know what they say, "angle and direction are the spice of life"! To find the angle, we can use the arctan function:

θ = arctan(a_tangential / a_centripetal)

Plugging in the values:

θ = arctan(39.7584 m/s^2 / 235.41568 m/s^2)
= arctan(0.169 m)

Using a calculator, we find:

θ ≈ 9.683 degrees

So, the angle of the total acceleration relative to the radial direction is approximately 9.683 degrees.

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To solve this problem, we can use the equations for centripetal acceleration and tangential acceleration. The total acceleration is the vector sum of these two accelerations.

Given:
Radius (r) = 0.608 m
Angular acceleration (α) = 65.3 rad/s²
Angular speed (ω) = 19.7 rad/s

(a) To find the magnitude of the total acceleration, we need to calculate the centripetal acceleration (ac) and tangential acceleration (at) separately, and then find their vector sum.

The formula for centripetal acceleration (ac) is:
ac = r * ω²

Substituting the given values, we have:
ac = (0.608 m) * (19.7 rad/s)²

Calculating this, we get:
ac ≈ 228.28 m/s²

The formula for tangential acceleration (at) is:
at = r * α

Substituting the given values, we have:
at = (0.608 m) * (65.3 rad/s²)

Calculating this, we get:
at ≈ 39.78 m/s²

The total acceleration (atotal) is the vector sum of ac and at:
atotal = √(ac² + at²)

Substituting the calculated values, we have:
atotal = √((228.28 m/s²)² + (39.78 m/s²)²)

Calculating this, we get:
atotal ≈ 233.05 m/s²

Therefore, the magnitude of the total acceleration is approximately 233.05 m/s².

(b) To determine the angle of the total acceleration relative to the radial direction, we can use the formula:

θ = arctan(at/ac)

Substituting the calculated values, we have:
θ = arctan(39.78 m/s² / 228.28 m/s²)

Calculating this, we get:
θ ≈ 9.97 degrees

Therefore, the angle of the total acceleration relative to the radial direction is approximately 9.97 degrees.

To find the magnitude of the total acceleration of the ball, we need to consider both the centripetal acceleration and the tangential acceleration.

(a) Magnitude of total acceleration (a_total):

The centripetal acceleration (a_c) is given by the formula:
a_c = r * ω^2
where r is the radius of the circle and ω is the angular speed.

Given:
r = 0.608 m
ω = 19.7 rad/s

Calculating centripetal acceleration (a_c):
a_c = (0.608 m) * (19.7 rad/s)^2
a_c ≈ 231.385 m/s^2

The tangential acceleration (a_t) can be calculated using the formula:
a_t = r * α
where α is the angular acceleration.

Given:
α = 65.3 rad/s^2

Calculating tangential acceleration (a_t):
a_t = (0.608 m) * (65.3 rad/s^2)
a_t ≈ 39.6784 m/s^2

To find the magnitude of the total acceleration (a_total):
a_total = √(a_c^2 + a_t^2)

Calculating magnitude of total acceleration (a_total):
a_total = √((231.385 m/s^2)^2 + (39.6784 m/s^2)^2)
a_total ≈ 235.758 m/s^2

Therefore, the magnitude of the total acceleration of the ball is approximately 235.758 m/s^2.

(b) Angle of total acceleration relative to the radial direction:

The angle can be found using the tangent of the angle:
tan(θ) = a_t / a_c

Calculating the angle (θ):
θ = tan^(-1)(a_t / a_c)

Calculating the angle (θ):
θ = tan^(-1)(39.6784 m/s^2 / 231.385 m/s^2)
θ ≈ 9.36 degrees

Therefore, the angle of the total acceleration relative to the radial direction is approximately 9.36 degrees.