A uniform solid sphere of mass M and radius R is rolling without sliding along a level plane with a speed v = 2.70 m/s when it encounters a ramp that is at an angle è = 20.4° above the horizontal. Find the maximum distance that the sphere travels up the ramp if the ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height.

Question

A uniform solid sphere of mass M and radius R is rolling without sliding along a level plane with a speed v = 2.70 m/s when it encounters a ramp that is at an angle è = 20.4° above the horizontal. Find the maximum distance that the sphere travels up the ramp if the ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height.

Answer 1

well, the translational KE converts to PE

1/2 m 2.7^2=mg*dsin20.4
solve for d

Answer 2

Thank you.

Answer 3

To find the maximum distance that the sphere travels up the ramp, we can use the principle of conservation of mechanical energy. The mechanical energy of the system remains constant throughout the motion, so we can equate the initial mechanical energy to the maximum potential energy.

The initial mechanical energy of the system consists of the kinetic energy of both translational and rotational motion.

1. Translational Kinetic Energy:
The translational kinetic energy of the rolling sphere is given by:
K_trans = 1/2 * M * v^2
where M is the mass of the sphere and v is the linear speed of the sphere.

2. Rotational Kinetic Energy:
The rotational kinetic energy of the rolling sphere is given by:
K_rot = 1/2 * I * ω^2
where I is the moment of inertia of the sphere and ω is its angular velocity.

For a solid sphere rolling without sliding, the relationship between linear and angular velocity is given by:
v = R * ω
where R is the radius of the sphere.

The moment of inertia of a solid sphere is given by:
I = (2/5) * M * R^2

Substituting the value of ω in terms of v and substituting the moment of inertia, we have:
K_rot = 1/2 * (2/5) * M * R^2 * (v/R)^2
K_rot = 1/5 * M * v^2

The total initial mechanical energy is the sum of the translational and rotational kinetic energies:
E_initial = K_trans + K_rot
E_initial = 1/2 * M * v^2 + 1/5 * M * v^2
E_initial = 7/10 * M * v^2

At the maximum height, all the initial kinetic energy is converted to potential energy.
So, the potential energy at the maximum height is given by:
PE_max = M * g * h_max
where g is the acceleration due to gravity and h_max is the maximum height achieved by the sphere.

Equating the initial mechanical energy to the potential energy at the maximum height:
E_initial = PE_max
7/10 * M * v^2 = M * g * h_max

Simplifying the equation, we get:
h_max = (7/10) * (v^2 / g)

Now, substituting the given values, we can calculate the maximum height h_max.