A bowling ball with a radius of 12 cm is given a backspin (i.e. if you picture the ball moving to the right it is spinning counterclockwise at the same time) of 8.3 rad/ s and a forward velocity of 8.5 m/s.As a result, the ball slides along the floor. What is the forward velocity of the point on the ball that is touching the floor?
answer i 9.5 m/s but how?
We can solve this problem by adding the forward linear velocity of the ball to the linear velocity due to the rotation of the ball.
First, we need to find the linear velocity due to the rotation of the ball. We know the angular velocity (8.3 rad/s) and the radius of the ball (12 cm, or 0.12 m).
The linear velocity v due to rotation can be found by the formula:
v = r * ω
where r is the radius of the ball and ω is the angular velocity.
v = 0.12 m * 8.3 rad/s = 0.996 m/s
Now, we add the forward linear velocity (8.5 m/s) and the linear velocity due to rotation (0.996 m/s).
Forward velocity of the point touching the floor = 8.5 m/s + 0.996 m/s ≈ 9.5 m/s
To find the forward velocity of the point on the ball that is touching the floor, you can use the concept of the velocity of a point on a rotating object.
The velocity of any point on a rotating object is the sum of the linear velocity due to the translational motion and the tangential velocity due to the rotational motion.
In this case, the ball is sliding along the floor, indicating translational motion, and it also has a backspin, indicating rotational motion.
The linear velocity due to translational motion is given as 8.5 m/s.
The tangential velocity due to rotational motion can be calculated using the formula:
v = ω * r
Where:
v is the tangential velocity,
ω is the angular velocity,
and r is the radius of the ball.
Given:
ω = 8.3 rad/s (angular velocity)
r = 12 cm = 0.12 m (radius)
Now, substitute the values to calculate the tangential velocity due to the backspin:
v = 8.3 rad/s * 0.12 m = 0.996 m/s
Since the ball is sliding along the floor, the forward velocity of the point touching the floor would be the sum of the linear velocity and the tangential velocity:
Forward velocity = Linear velocity + Tangential velocity
= 8.5 m/s + 0.996 m/s
= 9.496 m/s
Rounding to two decimal places, the forward velocity of the point on the ball touching the floor is approximately 9.50 m/s.
To find the forward velocity of the point on the ball that is touching the floor, we can use the concept of relative velocity.
First, let's break down the velocity of the ball into its two components: forward velocity and rotational velocity. The forward velocity of the ball is given as 8.5 m/s, while the rotational velocity, or backspin, is given as 8.3 rad/s.
The point on the ball that is touching the floor is not moving rotationally, as it is in contact with the stationary floor. However, it does have a forward velocity due to the overall motion of the ball.
To find the forward velocity of the point, we need to add the forward velocity of the ball to the linear velocity caused by the rotational motion at the point of contact.
The linear velocity caused by the rotational motion can be calculated using the formula:
v = ω * r
Where v is the linear velocity, ω (omega) is the rotational velocity, and r is the radius of the ball.
In this case, the radius of the ball is given as 12 cm, which is equal to 0.12 m. The rotational velocity is given as 8.3 rad/s.
So, the linear velocity caused by the rotational motion at the point of contact is:
v = 8.3 rad/s * 0.12 m = 0.996 m/s
Now, we can find the forward velocity of the point on the ball that is touching the floor by adding the forward velocity of the ball (8.5 m/s) to the linear velocity caused by the rotational motion (0.996 m/s).
Forward velocity of the point on the ball = Forward velocity of the ball + Linear velocity caused by rotational motion
= 8.5 m/s + 0.996 m/s
= 9.496 m/s
Therefore, the forward velocity of the point on the ball that is touching the floor is approximately 9.5 m/s.