What is the critical angle of inclination at which the hoop begins to slip?
Explain how the critical angle of inclination at which the hoop begins to slip is determined.
The critical angle of inclination at which the hoop begins to slip can be determined by considering the forces acting on the hoop. When the ramp is very steep, the hoop will experience a downward force due to gravity and an upward normal force from the ramp. In addition, there will be a static frictional force acting between the hoop and the ramp, preventing the hoop from slipping. To find the critical angle, we need to determine when the static frictional force reaches its maximum value before the hoop starts slipping. At this point, the static frictional force is equal to the product of the coefficient of static friction (µs) and the normal force. Let's assume the angle of inclination of the ramp is θ. The normal force can be calculated as the component of the weight of the hoop perpendicular to the ramp, which is given by N = mgcosθ, where m is the mass of the hoop and g is the acceleration due to gravity. The maximum static frictional force (fs) can then be calculated as fs = µsN = µsmgcosθ. Now, the critical angle of inclination is reached when the static frictional force fs reaches its maximum value, which is equal to the maximum static frictional force that can be exerted by the ramp on the hoop without slipping. The maximum static frictional force is also the force required to make the hoop start slipping, which can be calculated as the product of the normal force and the coefficient of kinetic friction (µk) between the hoop and the ramp. Therefore, fs = µkN = µkmgcosθ. Equating the two expressions for fs, we have µsmgcosθ = µkmgcosθ. Simplifying, we find µs = µk. Since µs represents the coefficient of static friction, and µk represents the coefficient of kinetic friction, we can conclude that the critical angle of inclination at which the hoop begins to slip is given by tanθ = 2µs. In conclusion, the critical angle of inclination at which the hoop begins to slip is given by tanθ = 2µs.