Question

(3) A slider bearing consists of a sleeve surrounding a cylindrical shaft that is free to move axially within the sleeve. A lubricant (e.g, grease) is in the gap between the sleeve and the shaft to isolate the metal surfaces and support the stress resulting from the shaft motion. The diameter of the shaft is 2.54 cm, and the sleeve has an inside diameter of 2.6 cm and a length of 5.08cm. If you want to limit the total force on the sleeve to less than 2.2 N when the shaft is moving at a velocity of 6.1 m/s, what should the viscosity of the grease be? What is the magnitude of the flux of momentum in the gap, and which direction is the momentum being transported?​

Answer 1

The force acting on the sleeve due to the shaft's motion can be calculated using the following formula:

F = 6πμvl

where F is the force, μ is the viscosity of the lubricant, v is the velocity of the shaft, and l is the length of the sleeve.

Plugging in the given values, we get:

2.2 N = 6πμ(6.1 m/s)(0.0508 m)

Solving for μ, we get:

μ = 0.041 Pa⋅s

Therefore, the viscosity of the grease should be 0.041 Pa⋅s to limit the total force on the sleeve to less than 2.2 N.

The flux of momentum in the gap can be calculated using the following formula:

Φ = πμr4v/l

where Φ is the flux of momentum, μ is the viscosity of the lubricant, r is the radius of the shaft, v is the velocity of the shaft, and l is the length of the sleeve.

Plugging in the given values, we get:

Φ = π(0.0254 m)4(0.041 Pa⋅s)(6.1 m/s)/0.0508 m

Φ = 0.047 N⋅s/m

The momentum is being transported in the direction of the shaft's motion, i.e., from the high-pressure end of the gap to the low-pressure end of the gap.