well, the translational KE converts to PE
1/2 m 2.7^2=mg*dsin20.4
solve for d
1/2 m 2.7^2=mg*dsin20.4
solve for d
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.