A solid cylinder of radius 0.85m is released from rest from a height of 1.8m and rolls down the incline as shown. What is the linear speed of the cylinder when it reaches the horizontal surface?
Total Ke = g m h = (1/2) m v^2 + (1/2) I w^2
where v = w r
so
m g h = (1/2) m v^2 + (1/2) (1/2) m r^2 (v^2/r^2)
2g h = v^2 + v^2/2 = 1.5 v^2
2 (9.81)(1.8) = 1.5 v^2
To find the linear speed of the cylinder when it reaches the horizontal surface, we can break down the problem into the following steps:
Step 1: Determine the potential energy at the initial position
The potential energy can be calculated using the formula: Potential Energy = mass * gravity * height. In this case, the height is given as 1.8m.
Step 2: Calculate the rotational energy at the initial position
The rotational energy of a rolling cylinder can be found using the formula: Rotational Energy = (1/2) * moment of inertia * angular velocity. The moment of inertia for a solid cylinder is given by the formula: moment of inertia = (1/2) * mass * radius^2.
Step 3: Equate the potential energy to the rotational energy at the initial position
Since the cylinder is released from rest, we can assume that the total mechanical energy is conserved. Therefore, at the top of the incline where the cylinder starts, the potential energy is equal to the rotational energy.
Step 4: Calculate the linear velocity at the bottom of the incline
Using the principle of conservation of mechanical energy, we can equate the initial potential energy to the sum of the final potential energy (at the lowest point) and the final kinetic energy. The kinetic energy of a rolling cylinder is given by: Kinetic Energy = (1/2) * moment of inertia * linear velocity^2.
Step 5: Solve for the linear velocity
By rearranging the equation from step 4 and solving for linear velocity, we can find the value of the linear speed of the cylinder when it reaches the horizontal surface.
Keep in mind that there might be additional factors to consider, such as friction or air resistance, which could affect the final result.