Rotational inertia of a sphere

In summary, the bowling ball has an initial speed of 8.5 m/s and initial angular speed of 0. The coefficient of kinetic friction is 0.21, causing linear and angular accelerations of -2.66 m/s^2 and 46.82 rad/s, respectively. To find the linear speed and time at which the ball begins to roll smoothly, equations for velocity in terms of time can be used, with the rotational inertia of the ball (2/5 mR^2) also being a factor. The answers to questions D, E, and F are all related to these variables.
  • #1
chaose
10
0
A bowler throws a bowling ball of radius R=0.11m along a lane. The ball slides on the lane with initial speed V of center of mass = 8.5 m/s and initial angular speed w = 0. The coefficient of kinetic friction between the ball and the lane is 0.21. The kinetic frictional force f acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed V of center of mass has decreased enough an angular speed w has increased enough, the ball stops sliding and then rolls smoothly.
A) What then is V center of mass in terms of w?

During the sliding, what are the ball's
B) linear acceleration and
C) angular acceleration?
D) how long does the ball slide?
E) how far does the ball slide?
F) what is the linear speed of the ball when smooth rolling begins?

The rotational inertia of a sphere is I = 2/5 mR^2.

What I have so far:

a) V center of mass = wR = 0.11w
b) f = -ma = umg, u = coefficient of kinetic friction.
a=-ug = -2.66m/s^2, negative because it's backwards. a here is linear acceleration.
c) torque = I*A = r x F, A = angular acceleration
2/5*(mR^2) A = rumg
A = (5/2R)*ug = 46.82 rad/s
 
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  • #2
chaose said:
A bowler throws a bowling ball of radius R=0.11m along a lane. The ball slides on the lane with initial speed V of center of mass = 8.5 m/s and initial angular speed w = 0. The coefficient of kinetic friction between the ball and the lane is 0.21. The kinetic frictional force f acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed V of center of mass has decreased enough an angular speed w has increased enough, the ball stops sliding and then rolls smoothly.
A) What then is V center of mass in terms of w?

During the sliding, what are the ball's
B) linear acceleration and
C) angular acceleration?
D) how long does the ball slide?
E) how far does the ball slide?
F) what is the linear speed of the ball when smooth rolling begins?

The rotational inertia of a sphere is I = 2/5 mR^2.

What I have so far:

a) V center of mass = wR = 0.11w
b) f = -ma = umg, u = coefficient of kinetic friction.
a=-ug = -2.66m/s^2, negative because it's backwards. a here is linear acceleration.
c) torque = I*A = r x F, A = angular acceleration
2/5*(mR^2) A = rumg
A = (5/2R)*ug = 46.82 rad/s
If you have both accelerations correct, you can write the equations for velocities in terms of time. At some time the "final" velocities will correspond to rolling. The velocity values and the time are three unknowns, but you have three equations. They can be solved for the velocities and the time. D, E, and F are all related to these variables.
 
  • #3
thanks. i didn't think of that
 

Related to Rotational inertia of a sphere

What is rotational inertia of a sphere?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. For a sphere, it is the product of its mass and the square of its radius.

How is rotational inertia different from linear inertia?

Rotational inertia is specific to an object's rotational motion, while linear inertia is specific to an object's linear motion. Rotational inertia depends on the distribution of mass within an object, while linear inertia depends on an object's mass and velocity.

How does the shape of a sphere affect its rotational inertia?

The shape of a sphere does not affect its rotational inertia, as long as the mass and radius remain constant. This is because a sphere has a symmetrical distribution of mass, resulting in the same moment of inertia regardless of its orientation.

What is the formula for calculating rotational inertia of a sphere?

The formula for calculating the rotational inertia of a solid sphere is I = 2/5 * m * r^2, where I is the moment of inertia, m is the mass of the sphere, and r is the radius of the sphere.

How does rotational inertia affect the motion of a sphere?

The higher the rotational inertia of a sphere, the more difficult it is to change its rotational motion. This means that a sphere with a higher moment of inertia will require more torque to accelerate or decelerate its rotation compared to a sphere with a lower moment of inertia.

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