A billiard ball that is initially at rest is given a sharp blow by a cue stick. The force is horizontal and is applied at a distance h = 2R/3 below the centerline, The speed of the ball just after the blow is v0 and the coefficient of kinetic friction between the ball and the billiard table is μk
find angular velocity just after blow, velocity once roles without slipping, and KE just after hit.
1.
The linear momentum is p =m•vₒ.
An angular momentum is
L = p•(2•R/3) = m•vₒ•(2•R/3).
Moment of inertia of the sphere is
I =2•m•R²/5.
L = I• ωₒ= (2•m•R²/5)•ωₒ.
m•vₒ•(2•R/3) = (2•m•R²/5)•ωₒ.
ωₒ = 5•vₒ/3•R.
Since the force is applied below the center line, the spin is
backward, i.e., the ball will slow down,
ωₒ = - 5•vₒ/3•R.
2.
M = F•R =μ•g•R is the torque of the friction force,
then an angular acceleration is
ε = M/I = μ•g•R/(2•m•R²/5) = 5• μ•g/2•R.
v =vₒ - a•t = vₒ- μ•g•t.
ω = ωₒ+ε•t= -(5•vₒ/3•R) +(5• μ•g/2•R)•t.
Since v =ω•R,
v = - (5•vₒ/3•R) •R +(5• μ•g/2•R) •R•t =
= - (5•vₒ/3)+ (5•μ•g/2) •t,
vₒ - μ•g•t = - (5•vₒ/3) + (5•μ•g/2) •t ,
vₒ + (5•vₒ/3) = μ•g•t + (5•μ•g/2) •t.
16•vₒ = 21•μ•g•t,
t = 16•vₒ/21•μ•g.
v = vₒ- μ•g•t = vₒ- μ•g•(16•vₒ/21•μ•g) =
= 5• 21 =0.238•vₒ.
3.
KE1 = m• vₒ²/2 + I•ωₒ²/2 =
= m• vₒ²/2 + (2•m•R²/5)•( 5•vₒ/3•R)²/2 =
=19• m• vₒ²/18 =1.056• m• vₒ².
KE2 = m• v²/2 + I•ω²/2 =
= m• v²/2 + (2•m•R²/5)•v²/2•R² = 0.7•m•v²=
0.7•m•(0.238•vₒ)²= 0.0397•m•vₒ².
ΔKE = - W(fr) = 1.056• m• vₒ² - 0.0397•m•vₒ² =
=1.016•m•vₒ².
Well, it seems like that billiard ball is in for a wild ride! Let's break it down, shall we?
To find the angular velocity just after the blow, we can use the principle of conservation of angular momentum. Since the ball is initially at rest, its initial angular momentum is zero. The angular momentum just after the blow will depend on the impulse applied by the cue stick.
Angular momentum (L) = Moment of inertia (I) * Angular velocity (ω)
Since the ball is given a sharp blow at a distance h below the centerline, we can calculate the impulse by multiplying the force (F) with the distance (h).
Impulse = F * h
Now, we can calculate the angular velocity just after the blow using the following equation:
L = I * ω
Since the ball is a solid sphere, the moment of inertia (I) is given by:
I = (2/5) * m * R^2
where m is the mass of the ball and R is its radius.
Great! Now, let's move on to finding the velocity once the ball rolls without slipping.
To calculate the velocity, we can use the relationship between linear velocity (v) and angular velocity (ω). For a ball rolling without slipping, the linear velocity is equal to the radius times the angular velocity, i.e.,
v = R * ω
For the final part of the question, we need to find the kinetic energy just after the hit. The kinetic energy (KE) is given by:
KE = (1/2) * m * v^2
where m is the mass of the ball and v is its linear velocity.
So, there you have it! To summarize:
1. To find the angular velocity just after the blow, use the principle of conservation of angular momentum.
2. To find the velocity once the ball rolls without slipping, use the relationship between linear and angular velocity.
3. To find the kinetic energy just after the hit, use the formula for kinetic energy.
I hope this helps you unravel the mysteries of that billiard ball's journey!
To find the angular velocity just after the blow, we can use the principle of conservation of angular momentum.
Step 1: Find the initial angular momentum of the ball.
The initial angular momentum (L_i) of the ball can be calculated using the equation:
L_i = I * ω_i,
where I is the moment of inertia of the ball and ω_i is the initial angular velocity.
Step 2: Find the final angular momentum of the ball.
After the blow, the ball will start rotating about its center of mass. The final angular momentum (L_f) can be calculated using the equation:
L_f = I * ω_f,
where ω_f is the final angular velocity.
Step 3: Apply the conservation of angular momentum.
According to the conservation of angular momentum principle, the initial angular momentum (L_i) should be equal to the final angular momentum (L_f):
L_i = L_f.
Step 4: Calculate ω_f.
Using the equation from Step 3, we can write:
L_i = I * ω_i = I * ω_f,
which gives:
ω_f = (I * ω_i) / I = ω_i.
Therefore, the angular velocity just after the blow is equal to the initial angular velocity: ω_f = ω_i.
To find the velocity once the ball rolls without slipping, we need to consider the forces acting on the ball.
Step 5: Calculate the net force acting on the ball.
The net force (F_net) acting on the ball can be determined by subtracting the force of kinetic friction (F_friction) from the applied force (F_applied):
F_net = F_applied - F_friction.
Step 6: Calculate the acceleration of the ball.
The acceleration (a) of the ball can be determined using Newton's second law:
F_net = m * a,
where m is the mass of the ball.
Step 7: Calculate the linear velocity once the ball rolls without slipping.
When the ball rolls without slipping, the linear velocity (v) can be related to the angular velocity (ω) and the radius (R) of the ball by the equation:
v = ω * R.
Therefore, the velocity once the ball rolls without slipping is v = ω_i * R.
To find the kinetic energy just after the hit, we need to consider the work done by the forces.
Step 8: Calculate the work done by the applied force.
The work done by the applied force can be calculated using the equation:
Work_applied = F_applied * d,
where d is the distance over which the force is applied.
Step 9: Calculate the work done by the force of kinetic friction.
The work done by the force of kinetic friction can be calculated using the equation:
Work_friction = F_friction * d.
Step 10: Calculate the net work done.
The net work done (Work_net) is the difference between the work done by the applied force and the work done by the force of kinetic friction:
Work_net = Work_applied - Work_friction.
Step 11: Calculate the kinetic energy just after the hit.
The kinetic energy just after the hit (KE) is equal to the net work done:
KE = Work_net.
By following these steps, you can find the angular velocity just after the blow, the velocity once the ball rolls without slipping, and the kinetic energy just after the hit.
To solve this problem, we'll use the principles of rotational motion and dynamics. Let's break down the problem step by step:
Step 1: Find the angular velocity just after the blow.
First, let's find the torque exerted by the force applied by the cue stick. The torque τ is given by:
τ = F * d
Where F is the force applied by the cue stick, and d is the perpendicular distance from the line of action of the force to the axis of rotation (center of the ball in this case).
Given that the force is applied at a distance h = 2R/3 below the centerline, the perpendicular distance is d = R/3.
Next, we know that torque is also related to the moment of inertia I and angular acceleration α:
τ = I * α
Since we're interested in finding the angular velocity ω just after the blow, we'll use the relation between angular velocity and angular acceleration:
α = Δω / Δt
Assuming that the angular acceleration is constant during the short time interval of the blow, we can rewrite the torque equation as:
F * d = I * Δω / Δt
Simplifying, we have:
I * Δω = F * d * Δt
Assuming the ball is a solid sphere, the moment of inertia I of a solid sphere is given by:
I = (2/5) * m * R²
Where m is the mass of the ball and R is its radius. Let's assume the mass of the ball is just m.
Substituting this into the equation, we get:
(2/5) * m * R² * Δω = F * d * Δt
We know that the force F is the impulse J divided by the time Δt, where the impulse J is the change in linear momentum of the ball:
J = m * Δv
Where Δv is the change in linear velocity of the ball. Since the ball was initially at rest, Δv = v0.
Substituting this into the equation, we get:
(2/5) * m * R² * Δω = (m * v0) * d
Now, we can solve for Δω:
Δω = (5/2) * v0 * d / R²
Finally, substituting the values of d = R/3, we find:
Δω = (5/2) * v0 * (R/3) / R² = (5/6) * v0 / R
So, the angular velocity just after the blow is (5/6) * v0 / R.
Step 2: Find the velocity once the ball rolls without slipping.
When a ball rolls without slipping, the linear velocity v and the angular velocity ω are related by:
v = R * ω
Substituting the value of ω that we found earlier, we get:
v = R * (5/6) * v0 / R = (5/6) * v0
So, the velocity of the ball once it starts rolling without slipping is (5/6) * v0.
Step 3: Find the kinetic energy just after the hit.
The formula for kinetic energy is:
KE = (1/2) * I * ω² + (1/2) * m * v²
Since the ball is rolling without slipping, the linear velocity v is equal to the velocity we found in step 2:
v = (5/6) * v0
Substituting the moment of inertia I of the solid sphere and the angular velocity ω that we found earlier:
KE = (1/2) * [(2/5) * m * R²] * [(5/6) * v0 / R]² + (1/2) * m * [(5/6) * v0]²
Simplifying, we get:
KE = (1/2) * (10/36) * m * v0² + (1/2) * (25/36) * m * v0²
KE = (5/18) * m * v0² + (25/72) * m * v0²
KE = (5/18 + 25/72) * m * v0²
KE = (35/72) * m * v0²
Therefore, the kinetic energy just after the hit is (35/72) * m * v0².