A solid sphere of mass 2.07kg rolls up an incline with an inclination angle of 24.40°. At the bottom of the incline, the center of mass of the sphere has a translational speed of 2.80m/s.
a) What is in J, the linear kinetic energy of the sphere at the bottom of the incline?
b) What is in J, the total kinetic energy of the sphere at the bottom of the incline?
c) How far up the incline will the sphere roll? Answer in units of m.
To solve this problem, we need to use the concepts of rotational and translational kinetic energy.
a) To find the linear kinetic energy of the sphere at the bottom of the incline, we'll use the formula for translational kinetic energy:
Kinetic energy = 1/2 * mass * velocity^2
Given:
Mass of the sphere, m = 2.07 kg
Translational speed of the sphere, v = 2.80 m/s
Plugging in the values into the formula:
Kinetic energy = 1/2 * 2.07 kg * (2.80 m/s)^2
Kinetic energy = 1/2 * 2.07 kg * 7.84 m^2/s^2
Evaluating the expression:
Kinetic energy ≈ 8.123 J
Therefore, the linear kinetic energy of the sphere at the bottom of the incline is approximately 8.123 J.
b) To find the total kinetic energy of the sphere at the bottom of the incline, we'll use the formula for total kinetic energy:
Total kinetic energy = Translational kinetic energy + Rotational kinetic energy
The rotational kinetic energy of a rolling sphere is expressed as:
Rotational kinetic energy = 1/2 * moment of inertia * angular velocity^2
For a solid sphere rolling without slipping, the moment of inertia is given by:
Moment of inertia (I) = 2/5 * mass * radius^2
Given:
Mass of the sphere, m = 2.07 kg
Translational speed of the sphere, v = 2.80 m/s
Radius of the sphere, r = ?
The radius is not given directly, but we can find it using the inclination angle of the incline. The relationship between the angle and the radius is given by:
sin(angle) = radius / height
Rearranging the equation to solve for the radius:
radius = sin(angle) * height
Plugging in the known values:
radius = sin(24.40°) * height
Now, to find the height of the incline, we know that at the bottom of the incline, the center of mass of the sphere has a translational speed of 2.80 m/s. So, we can use the conservation of mechanical energy to find the height of the incline:
Total energy at the bottom = Gravitational potential energy at the top + Total kinetic energy at the bottom
At the top of the incline, the gravitational potential energy is given by:
Gravitational potential energy = mass * gravity * height
Since the sphere rolls up the incline without slipping, its total kinetic energy is equal to the sum of translational and rotational kinetic energies:
Total kinetic energy at the bottom = Translational kinetic energy + Rotational kinetic energy
Setting up the equation and rearranging to solve for the height:
Total energy at the bottom = mass * gravity * height + Translational kinetic energy + Rotational kinetic energy
Plugging in the values:
1/2 * mass * velocity^2 = mass * gravity * height + Translational kinetic energy + Rotational kinetic energy
Solving for the height:
height = (1/2 * mass * velocity^2 - Translational kinetic energy - Rotational kinetic energy) / (mass * gravity)
Evaluating the expression:
height ≈ ((1/2 * 2.07 kg * (2.80 m/s)^2) - 8.123 J - (1/2 * (2/5 * mass * radius^2) * (v / radius)^2)) / (2.07 kg * 9.8 m/s^2)
After substituting the expression for radius:
height ≈ ((1/2 * 2.07 kg * (2.80 m/s)^2) - 8.123 J - (1/2 * (2/5 * 2.07 kg * (sin(24.40°) * height))^2) * (2.80 m/s / (sin(24.40°) * height))^2) / (2.07 kg * 9.8 m/s^2)
Since height appears on both sides of the equation, we'll have to solve this equation iteratively using numerical methods or a computer program.
c) The distance up the incline that the sphere will roll is equal to the height of the incline. We can find this using the value of height obtained in part b).