A spherical bowling ball with mass m = 4.1 kg and radius R = 0.116 m is thrown down the lane with an initial speed of v = 8.9 m/s. The coefficient of kinetic friction between the sliding ball and the ground is ¦Ì = 0.34. Once the ball begins to roll without slipping it moves with a constant velocity down the lane.

How long does it take the bowling ball to begin rolling without slipping?
4) How far does the bowling ball slide before it begins to roll without slipping?
5) What is the magnitude of the final velocity?
6) After the bowling ball begins to roll without slipping, compare the rotational and translational kinetic energy of the bowling ball:

v₀ = 8.9 m/s

m=4.1 kg
R = 0.116 m
μ=0.34
===
F=ma
F(fr)=μmg
F=F(fr)
ma= μmg
a= μg=0.34•9.8=3.33 m/s²
The torque M=Iε=(2mR²/5)•ε
M=F(fr)•R= μmgR
(2mR²/5)•ε= μmgR
ε= 5μmgR/2mR²= =5μg/2R=5•0.34•9.8/2•0.116=71.8 rad/s²
When speed v has decreased enough and angular speed ω has increased enough, the ball stops sliding and then rolls smoothly
v=v₀-at
ω= ω ₀+εt = εt
v= ωR
v₀-at= εtR

v₀ =t(a+εR)
t= v₀/(a+εR)=
=8.9/(3.33 +71.8•0.116)=0.76 s.

(4) s= at²/2=3.33•0.76²/2=0.962 m
(5) v=v₀-at=8.9 -3.33•0.76=6.37 m/s
(6) ω= εt = 71.8•0.76=54.57 rad/s

E(rot) = Iω²/2= 2mR²ω²/ 10=
=2•4.1•0.116²•54.57²/10=32.86 J.

E(tr) =mv²/2=4.1•6.37²/2=83.18 J

1) Well, that bowling ball sure needs a good push to start rolling without slipping! To find out how long it takes, we need to calculate the initial acceleration. The net force on the ball is the force of friction, which can be found using the coefficient of kinetic friction and the weight of the ball. So, F_friction = μ * m * g, where μ is the coefficient of kinetic friction, m is the mass of the ball, and g is the acceleration due to gravity.

Now, the frictional force provides the torque needed to get the ball rolling. And we know that torque is equal to the product of force and the radius. So, τ = F_friction * R.

For rolling motion, we also have the relation between the linear acceleration (a) and angular acceleration (α), which is given by α = a/R.

Since torque equals the product of moment of inertia (I) and angular acceleration (α), we can write τ = I * α.

For a solid sphere, the moment of inertia is given by I = (2/5) * m * R^2.

Now, let's put it all together. We have F_friction * R = I * α, which simplifies to μ * m * g * R = (2/5) * m * R^2 * α.

Simplifying further, we find α = (5/2) * μ * g * R.

The linear acceleration a is equal to the product of α and R, so a = α * R = (5/2) * μ * g * R^2.

Finally, using the kinematic equation v^2 = u^2 + 2as, where u is the initial velocity, v is the final velocity, and s is the distance traveled, we can solve for the time t at which the ball starts rolling without slipping.

Since the initial velocity is given as 8.9 m/s, and the final velocity when the ball starts rolling is 0 m/s (because it's not sliding anymore!), and the initial distance is 0 m, we have v^2 = u^2 + 2as, which simplifies to 0 = (8.9 m/s)^2 + 2 * a * s.

Plugging in the value of a from earlier, we can solve for s and find out how far the ball slides before rolling without slipping.

2) The distance (s) can be found using the kinematic equation v^2 = u^2 + 2as, where u is the initial velocity, v is the final velocity, a is the acceleration, and s is the distance traveled. In this case, since the ball starts from rest (u = 0 m/s), the equation simplifies to s = v^2 / (2a).

Substituting the given values, we have s = (8.9 m/s)^2 / (2 * a). Plugging in the value of a from earlier, we can calculate the distance the ball slides before rolling without slipping.

3) To find the magnitude of the final velocity (v), we can use the equation v^2 = u^2 + 2as, where u is the initial velocity (8.9 m/s), a is the acceleration (found earlier), and s is the distance traveled before rolling without slipping. Since the ball is rolling without slipping at the end, the final velocity is 0 m/s.

Plugging in the known values into the equation, we have 0 = (8.9 m/s)^2 + 2 * a * s. Rearranging the equation, we can solve for v.

4) Now it's time to compare the rotational and translational kinetic energy! The rotational kinetic energy (KE_rot) is given by KE_rot = (1/2) * I * ω^2, where I is the moment of inertia (found earlier) and ω is the angular velocity.

The translational kinetic energy (KE_trans) is given by KE_trans = (1/2) * m * v^2, where m is the mass and v is the velocity.

Since the ball is rolling without slipping, the angular velocity (ω) is related to the linear velocity (v) through the equation ω = v/R, where R is the radius of the ball.

Now, let's calculate the rotational and translational kinetic energy of the bowling ball and compare them!

To answer your questions step by step:

1) To determine how long it takes for the bowling ball to begin rolling without slipping, we need to consider the forces acting on it.
The main force in this case is the friction force between the ball and the ground. The ball will start rolling once the torque due to friction overcomes the torque exerted by the ball's inertia.

2) The torque due to friction can be calculated using the equation:
Torque = Radius * Force of friction (τ = R * F_friction)

3) The force of friction can be calculated using the equation:
Force of friction = Coefficient of friction * Normal force (F_friction = μ * N)
The normal force exerted on the ball is equal to its weight, which can be calculated as:
Weight = mass * gravity (N = m * g)

4) To find the distance the ball slides before it begins rolling without slipping, we can use the equation of motion in the horizontal direction:
Distance = initial velocity * time + 0.5 * acceleration * time^2
The acceleration in this case is the deceleration caused by friction, which can be calculated using the equation:
Acceleration = Force of friction / mass

5) The final velocity can be calculated as the velocity at which the ball is rolling without slipping. At this point, the kinetic energy due to translation and the kinetic energy due to rotation are equal.

6) The kinetic energy due to translation (KE_translational) can be calculated using the equation:
KE_translational = 0.5 * mass * velocity^2

The kinetic energy due to rotation (KE_rotational) can be calculated using the equation:
KE_rotational = 0.5 * moment of inertia * angular velocity^2

By comparing these two values, we can determine whether rotational or translational kinetic energy is greater.

To answer these questions, we will need to apply several principles of physics, including Newton's laws of motion and the conservation of energy.

1) To determine how long it takes for the bowling ball to begin rolling without slipping, we need to consider the force of friction acting on the ball. Initially, the ball slides, so the friction force acts opposite to the direction of the motion. Once the ball starts rolling without slipping, the static friction force acts as a torque and helps the ball to roll.

The equation that relates the friction force to the normal force and the coefficient of kinetic friction is F_friction = µ * N. Here, F_friction is the friction force, µ is the coefficient of kinetic friction, and N is the normal force.

The normal force, N, acting on the bowling ball is equal to the weight of the ball, which is given by N = m * g, where m is the mass and g is the acceleration due to gravity.

Initially, the net force acting on the ball is the difference between the gravitational force and the friction force: F_net = m * g - µ * N.

To determine the time it takes for the ball to begin rolling without slipping, we can use Newton's second law for rotational motion, which states that the net torque equals the moment of inertia of the object times its angular acceleration.

The moment of inertia for a solid sphere rolling without slipping is equal to (2/5) * m * R^2, where m is the mass of the sphere and R is its radius. The angular acceleration, α, is related to the linear acceleration, a, by the equation α = a / R.

Using these equations, we can set up the following equation:

(2/5) * m * R^2 * (a / R) = m * g - µ * N.

We can solve this equation to find the linear acceleration, a. Once we have the acceleration, we can use the equation v = u + a * t to find the time it takes for the ball to start rolling without slipping, where v is the final velocity, u is the initial velocity (in this case, u = 8.9 m/s), and t is the time.

2) To determine how far the bowling ball slides before it begins rolling without slipping, we can use the equation s = u * t + (1/2) * a * t^2, where s is the distance, u is the initial velocity, t is the time, and a is the linear acceleration (which we found in the previous step).

3) Once the ball starts rolling without slipping, it moves with a constant velocity down the lane. Therefore, the magnitude of the final velocity will be equal to the initial velocity (8.9 m/s in this case).

4) After the ball begins to roll without slipping, we can compare the rotational and translational kinetic energy. The translational kinetic energy is given by KE_trans = (1/2) * m * v^2, where m is the mass of the ball and v is its velocity. The rotational kinetic energy is given by KE_rot = (1/2) * I * ω^2, where I is the moment of inertia and ω is the angular velocity. Since the ball is rolling without slipping, we can relate the linear and angular velocities using v = ω * R, where R is the radius of the ball.