We will now examine how much time we need to stop a ship-engine-like cylinder. Suppose you take a rotating cylinder and let it roll to the right until it reaches a vertical wall. It then spins against the two walls as shown in the picture, slowly slowing down. How much time will it take in s for the cylinder to stop? The initial velocity of cylinder's surface when it touches the vertical wall is 30 m/s.
Its length is l=10 m, its radius is r=0.5 m and it's made out of wood of density ρ=800 kg/m3.
The coefficient of friction between the cylinder and the walls is μ=0.3.
To determine how much time it will take for the cylinder to stop, we need to consider the forces acting on the cylinder and apply Newton's second law of motion.
First, let's analyze the forces acting on the cylinder. When the cylinder touches the vertical wall, the force of friction acts in the opposite direction to its motion. The force of friction can be calculated using the equation:
Frictional Force = coefficient of friction * Normal Force
The normal force is equal to the weight of the cylinder, which is given by:
Weight = Mass * Gravity
The mass of the cylinder can be calculated using its density and volume, where:
Mass = Density * Volume
The volume of the cylinder is given by:
Volume = π * (radius^2) * length
Once we have determined the mass of the cylinder and the frictional force, we can apply Newton's second law of motion:
Force = Mass * Acceleration
In this case, the force acting on the cylinder is the frictional force, and the acceleration is the negative of the initial velocity divided by the time it takes to stop. Rearranging the equation, we have:
Time = -Initial Velocity / (Force / Mass)
Now let's substitute the given values into the equations to calculate the time it takes for the cylinder to stop:
Density (ρ) = 800 kg/m^3
Radius (r) = 0.5 m
Length (l) = 10 m
Coefficient of friction (μ) = 0.3
Initial Velocity = 30 m/s
Acceleration (a) = -Initial Velocity / Time
Calculating the volume:
Volume = π * (0.5^2) * 10 = 7.85 m^3
Calculating the mass:
Mass = Density * Volume = 800 * 7.85 = 6280 kg
Calculating the frictional force:
Weight = Mass * Gravity = 6280 * 9.8 = 61484 N
Frictional Force = Coefficient of friction * Normal Force = 0.3 * 61484 = 18445.2 N
Finally, we can calculate the time it takes for the cylinder to stop:
Time = -Initial Velocity / (Force / Mass) = -30 / (18445.2 / 6280) ≈ -30 / 2.94 ≈ -10.2 s
Since time cannot be negative in this context, we take the absolute value of the time, giving us an approximate answer of 10.2 seconds for the cylinder to stop.