Answer :
To determine the forces related to the sliding of the lathe tail stock, we need to use the concepts of static and kinetic friction. Here's how we can solve this problem step-by-step:
Understand the Forces Involved:
- Static Friction ([tex]f_s[/tex]): This is the frictional force that must be overcome to start the movement. It can be calculated as [tex]f_s = \mu_s \cdot N[/tex], where [tex]\mu_s[/tex] is the coefficient of static friction, and [tex]N[/tex] is the normal force.
- Kinetic Friction ([tex]f_k[/tex]): Once the object is moving, kinetic friction comes into play. It is calculated as [tex]f_k = \mu_k \cdot N[/tex], where [tex]\mu_k[/tex] is the coefficient of kinetic friction.
Calculate the Normal Force ([tex]N[/tex]):
- The normal force is the force exerted by a surface to support the weight of an object resting on it, acting perpendicular to the surface. For a lathe tail stock resting on a horizontal surface, the normal force equals the weight of the tail stock.
- Weight ([tex]W[/tex]) = mass ([tex]m[/tex]) [tex]\times[/tex] gravity ([tex]g = 9.8 \, \text{m/s}^2[/tex]).
- [tex]N = m \cdot g = 12 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 117.6 \, \text{N}[/tex].
Calculate the Maximum Static Friction:
- [tex]f_s = \mu_s \cdot N = 0.3 \cdot 117.6 \, \text{N} = 35.28 \, \text{N}[/tex].
- This means the maximum horizontal force that can be applied without causing it to slide is 35.28 N.
Calculate the Kinetic Friction:
- [tex]f_k = \mu_k \cdot N = 0.2 \cdot 117.6 \, \text{N} = 23.52 \, \text{N}[/tex].
- This is the force required to keep the tail stock moving once it has started sliding.
In summary, the maximum horizontal force that can be applied without causing the lathe tail stock to slide is 35.28 N. Once it starts moving, a minimum horizontal force of 23.52 N is required to keep it moving along the lathe bed. This results from overcoming the kinetic friction.
