new package moving system in the new improved post office consists of a large circular disc which rotates once every 5 seconds at a constant speed in a horizontal plane. packages are put on outer edge of turntable on one side of room and removed on opposite side. coefficient of static friction between surface and package is .65. if system is to work what is maximum possible radius of turntable
To determine the maximum possible radius of the turntable, we need to consider the relationship between the coefficient of static friction, the rotation speed of the disc, and the maximum centripetal force that can be provided by the friction between the surface and the package.
Here's the step-by-step process to find the solution:
1. Recall the formula for centripetal force: F = m * R * ω^2, where F is the centripetal force, m is the mass of the package, R is the radius of the turntable, and ω (omega) is the angular velocity.
2. At the maximum radius, the static friction force will be equal to the maximum centripetal force needed to keep the package in circular motion. Therefore, F_friction = F_max = m * R * ω^2.
3. The maximum static friction force can be calculated using the formula: F_friction = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force acting on the package. In this case, the normal force is equal to the weight of the package, which is given by m * g, where g is the acceleration due to gravity.
4. Combining the above equations, we get: μ_s * N = m * R * ω^2.
5. Substitute in the equation for the normal force: μ_s * m * g = m * R * ω^2.
6. Simplify the equation: μ_s * g = R * ω^2.
7. Rearrange the equation to solve for R: R = (μ_s * g) / ω^2.
8. Plug in the given values for the coefficient of static friction (μ_s = 0.65) and the angular velocity (ω = 2π / T, where T is the time for one revolution which is 5 seconds): R = (0.65 * g) / (2π / 5)^2.
9. Calculate the maximum possible radius of the turntable using the value of acceleration due to gravity (g ≈ 9.8 m/s^2): R ≈ (0.65 * 9.8) / (2π / 5)^2.
10. Solve for R to find the maximum possible radius of the turntable.
After performing the calculations, the maximum possible radius of the turntable is approximately 1.146 meters.