A can of juice rolls down a ramp with angle 80 degrees. If the slope were frictionless the can would slide down the incline without rolling. Calculate the minimum coefficient of static friction needed for the can to roll down the incline. Model the can as a cylinder of mass 500 g, radius 10 cm and length 32 cm. Hint: Use Newton's Second Law
To calculate the minimum coefficient of static friction needed for the can to roll down the incline, we can use Newton's Second Law.
First, let's analyze the forces acting on the can:
1. The force of gravity (mg): This force acts straight downward. The magnitude of this force can be calculated as the product of the mass (m) and the acceleration due to gravity (g). In this case, the mass of the can is given as 500 g, which is equal to 0.5 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
So, the force of gravity is given by F_gravity = mg = 0.5 kg * 9.8 m/s^2 = 4.9 N.
2. The normal force (N): This force acts perpendicular to the incline and is equal in magnitude but opposite in direction to the vertical component of the force of gravity. The normal force counteracts the force of gravity and prevents the can from sinking into the incline.
3. The static friction force (f_s): This force acts parallel to the incline and opposes the motion of the can. For the can to roll down the incline, the friction force should be sufficient to provide the necessary torque for rotation.
Since the incline is frictionless, the static friction force should provide the necessary torque for the can to roll. The torque due to the static friction force can be given by the product of the static friction force (f_s) and the radius (r).
Now, we can derive the minimum coefficient of static friction needed for the can to roll down the incline:
The net torque acting on the can can be given by the difference in torques produced by the forces acting on the can. In this case, the difference in torque is due to the static friction force and has the magnitude equal to the torque required for rolling.
The torque required for rolling can be calculated as the product of the force of gravity (mg) and the radius (r). So, the torque required for rolling is given by τ_roll = mg * r.
The torque produced by the static friction force is given by τ_static_friction = f_s * r.
To derive the minimum coefficient of static friction, we set these torques equal to each other:
τ_roll = τ_static_friction
mg * r = f_s * r
Cancelling out the radius (r) on both sides, we get:
mg = f_s
Now, substituting the values, we have:
f_s = 4.9 N
Therefore, the minimum coefficient of static friction needed for the can to roll down the incline is equal to the magnitude of the force of gravity acting on the can, which is 4.9 N.