A solid cylinder has a mass of M and radius R. If it rolls without slipping on a horizontal surface with a linear speed of v, how high will it roll up an incline of angle θ? Don’t worry about friction. The moment of inertia for a cylinder is (1/2)MR2. Express the vertical height in terms of the given variables M, R, v, θ and g (some might not be used).
To find the vertical height that the solid cylinder will reach while rolling up the incline, we can use the principle of conservation of mechanical energy.
First, we need to determine the initial kinetic energy and potential energy of the cylinder at the bottom of the incline.
The initial kinetic energy (K1) of the cylinder is given by:
K1 = (1/2) * M * v^2
The initial potential energy (U1) is zero since the cylinder is at the bottom of the incline.
Next, we need to determine the final kinetic energy and potential energy of the cylinder at the highest point it reaches on the incline.
The final kinetic energy (K2) of the cylinder is zero since it reaches its highest point and comes to a stop momentarily.
The final potential energy (U2) is given by:
U2 = M * g * h
where h is the vertical height we want to find.
According to the principle of conservation of mechanical energy, the initial mechanical energy (K1 + U1) is equal to the final mechanical energy (K2 + U2).
Therefore, we have:
(1/2) * M * v^2 + 0 = 0 + M * g * h
Simplifying the equation:
M * v^2 = 2 * M * g * h
Dividing both sides of the equation by M and simplifying further:
v^2 = 2 * g * h
We can rearrange the equation to isolate h:
h = (v^2) / (2 * g)
So the vertical height the solid cylinder will reach on the incline is given by:
h = (v^2) / (2 * g)
where v is the linear speed of the cylinder, and g is the acceleration due to gravity.
Note: The radius of the cylinder (R) and the angle of the incline (θ) do not affect the calculation because we are assuming there is no friction.
its initial KE = 1/2 Mv^2
all that has to go into PE, Mgh
so, gh = 1/2 v^2
and the distance rolled is h/sinθ, right?
See where that takes you.