To calculate the angular velocity of the sphere at the bottom of the incline, we can use the conservation of energy and the relationship between linear velocity and angular velocity for a rolling sphere.
a) Angular velocity at the bottom of the incline:
The potential energy at the top of the incline is converted into both translational kinetic energy and rotational kinetic energy at the bottom. The conservation of energy equation can be written as:
m * g * h = (1/2) * m * v^2 + (1/2) * I * ω^2
where:
m is the mass of the sphere,
g is the acceleration due to gravity (approximately 9.8 m/s²),
h is the height of the incline,
v is the linear velocity of the sphere,
I is the moment of inertia of the sphere,
ω is the angular velocity of the sphere.
The moment of inertia of a solid sphere about its diameter can be calculated as:
I = (2/5) * m * r^2
where:
r is the radius of the sphere.
Given:
Diameter of the sphere = 7.70 cm = 0.077 m (which gives a radius of 0.0385 m)
Mass of the sphere = 380 g = 0.38 kg
Height of the incline, h = 1.70 m
Substituting the values into the equation, we have:
m * g * h = (1/2) * m * v^2 + (1/2) * (2/5) * m * r^2 * ω^2
Canceling out the mass and simplifying the equation:
g * h = (1/2) * v^2 + (1/5) * r^2 * ω^2
Solving for ω:
ω = sqrt((5 * (g * h - (1/2) * v^2)) / (r^2))
Substituting the known values:
ω = sqrt((5 * (9.8 m/s^2 * 1.70 m - (1/2) * (v = 0) m/s^2)) / (0.0385 m)^2)
Simplifying:
ω = sqrt((5 * (9.8 m^2/s^2 * 1.70 m)) / (0.0385 m)^2)
Calculating the result:
ω ≈ 10.497 rad/s
Therefore, the angular velocity of the sphere at the bottom of the incline is approximately 10.497 rad/s.
b) Fraction of kinetic energy that is rotational:
The fraction of kinetic energy that is rotational can be calculated using the equation:
Fraction of rotational kinetic energy = (1/2) * I * ω^2 / (1/2) * m * v^2
Canceling out the common terms:
Fraction of rotational kinetic energy = I * ω^2 / (m * v^2)
Substituting the known values:
Fraction of rotational kinetic energy = ((2/5) * m * r^2) * ω^2 / (m * v^2)
Simplifying:
Fraction of rotational kinetic energy = (2/5) * (r^2 * ω^2) / v^2
Substituting the known values:
Fraction of rotational kinetic energy = (2/5) * ((0.0385 m)^2 * (10.497 rad/s)^2) / (0.38 kg * 0 m/s)^2
Calculating the result:
Fraction of rotational kinetic energy ≈ 0.265
Therefore, approximately 26.5% of