A solid cylinder with an unknown radius starts from rest at the top of a 12m long ramp inclined 20.3 degrees above the horizontal. Calculate the cylinder's final velocity when it reaches the bottom of the ramp.
To solve this problem, we need to consider the conservation of energy.
The potential energy at the top of the ramp is given by:
PE_top = m * g * h
where m is the mass of the cylinder, g is the acceleration due to gravity, and h is the height of the ramp.
The kinetic energy at the bottom of the ramp is given by:
KE_bottom = (1/2) * m * v^2
where v is the final velocity of the cylinder.
Since the cylinder starts from rest, its initial kinetic energy is zero.
The total mechanical energy of the system is conserved, so we can write:
PE_top = KE_bottom
m * g * h = (1/2) * m * v^2
Simplifying, we get:
v^2 = 2 * g * h
v = sqrt(2 * g * h)
Now, let's calculate the values of g and h.
The acceleration due to gravity, g, is approximately 9.8 m/s^2.
The height of the ramp, h, can be calculated using the formula:
h = length_of_ramp * sin(angle_of_inclination)
h = 12m * sin(20.3 degrees)
h ≈ 4.17 m
Substituting the values into the equation for v:
v = sqrt(2 * 9.8 m/s^2 * 4.17 m)
v ≈ 8.59 m/s
Therefore, the final velocity of the cylinder when it reaches the bottom of the ramp is approximately 8.59 m/s.