Newton's second law and the kinematics of rotation allow us to find the angular velocity of the snowball as it rolls across the roof is:
w = [tex]\frac{2g}{R}[/tex]
The kinematics of rotational motion studies the rotational motion of bodies.
w = w₀ + 2 α θ
Where w and w₀ are the current and initial angular velocities, α the angular acceleration and θ the angle traveled.
Newton's second law establishes a relationship between the force, mass, and acceleration of the body.
The linear and rotational moviments are related.
a = α R
Where a and α are the linear and rotational accelerations, respectively, and R is the radius of the body.
Let's find the linear acceleration of the body, in the attached we see a diagram of the forces, let's use trigonometry to decompose the weight.
[tex]sin \theta = \frac{W_x}{W}[/tex]
Wₓ = W sin θ
Wₓ = m a
mg sin θ = m a
a = g sin θ
Now we can find the angular acceleration.
α = a / R
α = [tex]\frac{g}{ R \ sin \theta }[/tex]
The body is released therefore its initial velocity is zero, we substitute in the kinematics expression.
[tex]w = 2 ( \frac{g}{R \ sin \theta }) \ \theta[/tex]
in rotational motion the angles are measured in radians We use trigonometry to find the relationship between the angle and the distance traveled
[tex]\theta = \frac{h}{D}[/tex]
Where h is the height of the ceiling and D is the distance traveled. Let's substitute.
[tex]w = 2 \frac{g}{R sin \theta } \frac{h}{D}[/tex]
Let's tirgonmetry.
sin θ = [tex]\frac{h}{D}[/tex]
w = [tex]\frac{2g}{R}[/tex]
In conclusion, using Newton's second law and the kinematics of rotation, we can find the angular velocity of the snowball when rolling on the roof is:
w = [tex]\frac{2g}{R}[/tex]
Learn more here: brainly.com/question/14455108
