if the speed of the ball is 145km/h and the parimeter of the ball is 22.4cm along the seam and spin ratio is 0.1 find angular speed approximately imparted to the ball.

To find the angular speed imparted to the ball, we can use the relationship between linear speed, angular speed, and the radius of the ball. The formula we will use is:

Understanding the Relationship

The linear speed (v) of an object is related to its angular speed (ω) by the equation:

v = ω * r

Where:

• v is the linear speed in meters per second (m/s).
• ω is the angular speed in radians per second (rad/s).
• r is the radius of the ball in meters (m).

Converting Units

First, we need to convert the speed of the ball from kilometers per hour (km/h) to meters per second (m/s). The conversion factor is:

1 km/h = 1/3.6 m/s

So, for a speed of 145 km/h:

v = 145 km/h * (1/3.6 m/s per km/h) ≈ 40.28 m/s

Finding the Radius

Next, we need to determine the radius of the ball. The perimeter (circumference) of the ball is given as 22.4 cm. The relationship between the circumference (C) and the radius (r) is:

C = 2 * π * r

Rearranging this to find the radius gives us:

r = C / (2 * π)

Substituting the given circumference:

r = 22.4 cm / (2 * π) ≈ 3.57 cm

Now, converting this to meters:

r ≈ 0.0357 m

Calculating Angular Speed

Now that we have both the linear speed and the radius, we can find the angular speed using the rearranged formula:

ω = v / r

Substituting the values we have:

ω = 40.28 m/s / 0.0357 m ≈ 1126.5 rad/s

Considering the Spin Ratio

The spin ratio of 0.1 indicates how much of the linear speed contributes to the spin of the ball. To find the effective angular speed imparted to the ball, we multiply the calculated angular speed by the spin ratio:

Effective Angular Speed = ω * Spin Ratio

Effective Angular Speed ≈ 1126.5 rad/s * 0.1 ≈ 112.65 rad/s

Final Result

Thus, the approximate angular speed imparted to the ball is about 112.65 rad/s. This value reflects how quickly the ball is spinning as it travels through the air, influenced by both its linear speed and the spin ratio.