A solid sphere of radius 35 cm is positioned
at the top of an incline that makes 20 ◦
angle
with the horizontal. This initial position of
the sphere is a vertical distance 3.4 m above
its position when at the bottom of the incline.
The sphere is released and moves down the
incline. 35 cm
M
µ
ℓ
20 ◦
3.4 m
Find the speed of the sphere when it reaches
the bottom of the incline if it rolls without
slipping. The acceleration of gravity is
9.8 m/s
2
and the moment of inertia of a sphere
with respect to an axis through its center is
2
5
M R2
.
Answer in units of m/s.
To find the speed of the sphere when it reaches the bottom of the incline, we can use the principles of energy conservation.
First, let's calculate the initial potential energy of the sphere when it is positioned at the top of the incline. The potential energy is given by the equation:
PE = m * g * h
Where m is the mass of the sphere, g is the acceleration due to gravity, and h is the vertical distance above the bottom of the incline.
In this case, we are given the radius of the sphere (35 cm) but not the mass. However, we can use the density of the material the sphere is made of to calculate the mass.
Density of sphere = mass / volume
The volume of a sphere is given by the formula:
volume = (4/3) * pi * R^3
Where R is the radius of the sphere.
Now, we can calculate the mass:
volume * density = mass
Next, we can calculate the initial potential energy:
PE = mass * g * h
Now, let's calculate the final kinetic energy of the sphere when it reaches the bottom of the incline. The kinetic energy is given by the equation:
KE = (1/2) * I * w^2
Where I is the moment of inertia of the sphere and w is the angular velocity. In this case, since the sphere is rolling without slipping, the angular velocity is related to the linear velocity (v) of the sphere by the equation:
w = v / R
So we can rewrite the equation for kinetic energy as:
KE = (1/2) * I * (v/R)^2
Finally, since there is no loss of energy in this system, we can equate the initial potential energy to the final kinetic energy:
PE = KE
Solving for v, we can find the speed of the sphere when it reaches the bottom of the incline.