A solid cylinder with an unknown radius starts from rest at the top of a 12.0 m long ramp inclined 20.3 degrees above the horizontal. Calculate the cylinder's final velocity when it reaches the bottom of the ramp.
To find the final velocity of the cylinder when it reaches the bottom of the ramp, we can use the principles of conservation of energy.
The potential energy at the top of the ramp is converted into kinetic energy at the bottom of the ramp. The potential energy at the top is given by:
PE_initial = mgh
where m is the mass of the cylinder, g is the acceleration due to gravity (9.81 m/s²), and h is the vertical height of the ramp. Since the cylinder is a solid object, we can use the formula for the cylinder's volume to find its mass:
V = πr²h
where r is the radius of the cylinder.
Given that the ramp is 12.0 m long and inclined at an angle of 20.3 degrees, the vertical height h can be found using trigonometry:
h = 12.0 sin(20.3°) = 4.39 m
Now, we can find the initial potential energy of the cylinder:
PE_initial = mgh = ρVgh = ρ(πr²h)g
Next, we find the kinetic energy of the cylinder at the bottom of the ramp:
KE_final = 0.5mv²
By the principle of conservation of energy, the initial potential energy is equal to the final kinetic energy:
PE_initial = KE_final
ρ(πr²h)g = 0.5mv²
We can solve this equation to find the final velocity v:
ρ(πr²h)g = 0.5mv²
ρ(πr²h)g = 0.5ρVv²
πr²gh = 0.5πr²hv²
2gh = v²
v = √(2gh) = √(2(9.81 m/s²)(4.39 m))
v ≈ 9.9 m/s
Therefore, the final velocity of the cylinder when it reaches the bottom of the ramp is approximately 9.9 m/s.