Final answer:
To find the linear speed of the cylinder when it reaches the bottom of the ramp, we can use the principle of conservation of energy. The gravitational potential energy at the top of the ramp will be converted into rotational kinetic energy at the bottom. By setting the initial potential energy equal to the rotational kinetic energy, we can calculate the linear speed of the cylinder.
Step-by-step explanation:
To find the linear speed of the cylinder when it reaches the bottom of the ramp, we can use the principle of conservation of energy. The initial potential energy at the top of the ramp is converted into kinetic energy at the bottom.
First, we need to find the gravitational potential energy at the top of the ramp:
PE = mgh = (3.0 kg)(9.8 m/s^2)(1.5 m) = 44.1 J
Next, we can find the rotational kinetic energy of the rolling cylinder at the bottom of the ramp:
KE_rot = 1/2 Iω^2 = 1/2 (MR^2)(v/R)^2 = 1/2 (3.0 kg)(0.2 m)^2 (v/0.2 m)^2,
where I is the moment of inertia of a solid cylinder and ω is the angular velocity. Since the cylinder rolls without slipping, ω = v/R.
Finally, we can set the initial potential energy equal to the rotational kinetic energy and solve for the linear speed v:
PE = KE_rot => 44.1 J = 1/2 (3.0 kg)(0.2 m)^2 (v/0.2 m)^2
Simplifying and solving for v:
v = √(2(44.1 J) / (3.0 kg)(0.2 m)^2)
Calculating the value of v gives us the linear speed of the cylinder when it reaches the bottom of the ramp.