In a popular amusement park ride, a cylinder of radius 2.19 meters is set in rotation at an angular speed of 5.09 rad/s. The floor then drops away, leaving the riders suspended against the wall in a vertical position. What minimum coefficient of friction between a rider's clothing and the wall of the cylinder is needed to keep the rider from slipping?
I am so lost on this problem. How do I solve this?
To solve this problem, we need to analyze the forces acting on the rider to determine the minimum coefficient of friction needed to keep them from slipping.
Let's break down the problem step by step:
1. Identify the forces:
- The weight of the rider acts downward, towards the center of the Earth.
- The normal force acts perpendicular to the surface of the wall.
- The frictional force acts tangentially along the surface of the wall.
2. Determine the centripetal force:
- The rider is moving in a circular path due to the rotation of the cylinder, which requires a centripetal force. In this case, the centripetal force is provided by the frictional force between the rider's clothing and the wall of the cylinder.
3. Use the centripetal force equation:
- The centripetal force is given by the product of the mass (m) of the rider, the radius (r) of the cylinder, and the angular speed (ω) squared: F_c = m * r * ω^2.
4. Equate the centripetal force to the maximum static friction:
- Since the riders are at the verge of slipping, the static frictional force will be at its maximum. Thus, we can equate the centripetal force to the maximum static friction force: F_c = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force.
5. Determine the normal force:
- The normal force is equal to the weight of the rider when they are in a vertical position against the wall: N = m * g.
6. Substitute the values into the equations:
- Substitute the known values for the radius (r), angular speed (ω), and coefficient of friction (μ_s) into the equation and solve for the minimum coefficient of friction that keeps the rider from slipping.
Remember to convert any units to the appropriate SI units (meters, kilograms, seconds, etc.) before plugging them into the equations.
By following these steps, you should be able to solve the problem and determine the minimum coefficient of friction needed to keep the rider from slipping.