_{1}

^{*}

To demonstrate the influence of sliding friction of motion on a curve, a circular path is considered for simplicity on which a person slides from the highest point to the lowest point. A slide which represents a quadrant of radius 5 m and a person of mass 60 kg are considered for comparison in this paper. A Differential equation for motion considering the fact that the normal force depends both on the sin component of weight and also on the tangential velocity, is established and is solved using integrating factor method, and the motion is analysed for different surface roughness of the slide and is compared using superimposed graphs, also the limiting value of friction coefficient at which the person just exits the slide is determined. The correction factor for exit velocity with friction as compared with the exit velocity for zero friction is determined. The fraction of energy lost to friction at the exit is evaluated. The Variation of normal force with the position of the person on the slide is plotted for different surface roughness of the slide, and the position on the slide where the normal force or the force experienced by the person is maximum, is determined and hence its maximum value is evaluated for different surface roughness. For simplicity, a point contact between the body and the slide is considered.

Friction put in simple words is the resistance to motion because of the contact between the body and the supporting surface. It is basically classified into two types―static and dynamic friction, static friction acts between two bodies with no relative motion with respect to each other whereas dynamic friction acts between two bodies with a relative motion with respect to each other, several attempts have been made to understand the general working of friction [

Ø First a slide which represents a quadrant of radius 5 m (for simplicity) is assumed on which the person of mass m = 60 kg slides from the top(q = 0) to the exit or the end of the slide (q = 90 degree) and the differential equation for motion is established and is solved using integrating factor method to get the variation of tangential velocity (v) with the position (q) on the slide

Ø Exit velocity(the velocity at which the person exits the slide) is determined as a function of friction coefficient (u) and is compared with the exit velocity for zero friction and a term called ‘the correction factor’ is defined which accounts for the change in exit velocity due to the presence of friction ,and another term called “critical coefficient of friction” is defined which is a limiting value of friction coefficient beyond which the person ceases to exit or come out of the slide

Ø Motion is analyzed for different surface roughness (different values of u) of the slide and a superimposed graph of v vs q is plotted to clearly show the effects of different surface roughness on the velocity profile

Ø The variation of the normal force on the person with his position on the slide is plotted and the position where the normal force is maximum and hence its value is determined for different surface roughness values of the slide

Ø Percentage of energy lost to friction is determined as a function of coefficient of friction u and a graph is plotted between the two

By referring to

m = mass of the person (60 kg)

q = angle of inclination with the horizontal

N = normal force

f = frictional force

u = coefficient of friction

g = acceleration due to gravity (9.81 m/s^{2})

v = absolute velocity of the person

R = Radius of curvature of the slide (5 m)

Retarding force = friction force =

N = fn (cos component of weight, centrifugal force)

From FBD, by equilibrium of forces

But

But

Multiplying by R and dividing by m we get:

Thus Equation (5) represents the differential equation of motion on the slide

Now if we put u = 0 the equation becomes

which represents the motion without friction which upon solving gives

Solution:

The Equation (5) represents non-linear single order ordinary differential equation:

To solve put v^{2} = z

Differentiating w.r.t q,

Thus substituting in Equation (1) we get:

Now this type of differential equation can be solved by integrating factor method:

Integrating factor

Thus the solution will be:

But z = v^{2} thus substituting:

Now exit velocity is obtained by setting q = 90˚ or π/2 rad:

Now exit velocity without friction can be obtained by setting u=0

Thus

Thus to consider friction effects, a correction factor f is to be multiplied,

Fraction of Energy lost to friction:

Determination of maximum normal force on the person and the location on the track (q_{n}) where it occurs:

Now to find the maximum value of N differential of N, w r t q s

Now to find the maximum value of N differential of N, w r t q should be zero

Thus

Thus by solving the following equation:

For different values of u, different positions (q_{n}) values for which the normal force is maximum are obtained

Considering a case where R = 5 m, exit velocity vs u graph is plotted.

It can be inferred from _{exit} = 9.9 m/s _{c}_{ }

which gives:

After solving u_{c} is obtained as 0.60259, thus for all practical purposes, the person comes out only if the roughness combination of his clothing and the slide surface produces a friction coefficient of less than 0.60259 (for all u < u_{c}) A table below lists the values of exit velocity for different values of u (R = 5 m) it is to be noted that u_{c} does not depend on any initial condition and is a characteristic of the geometry under consideration (circular surface in this case) hence u_{c} for a circular surface is found to be 0.60259

This gives the variation of absolute velocity of the person with his angular position (q) on the slide, the curves are plotted for different values of friction coefficient u of the slide surface and the graphs are compared by: superimposing

It can be inferred from the _{c} (critical coefficient of friction) the curve touches the x axis thus v_{exit} will become zero and it will remain zero (the person will not come out) for all values of u > u_{c}.

u | v_{exit} (m/s) |
---|---|

0 | 9.904 |

0.1 | 7.246 |

0.2 | 5.112 |

0.3 | 4.264 |

0.4 | 2.890 |

0.5 | 0.932 |

0.60259 | 0 |

It can be inferred from _{c} (critical value). _{max} for different values of u and the angle at which it occurs.

q_{n} (Degree) | N_{max} (N) | u |
---|---|---|

90 | 1765.58 | 0 |

80.39 | 1474.25 | 0.1 |

73.31 | 1268 | 0.2 |

67.6 | 1113.47 | 0.3 |

63.28 | 993 | 0.4 |

59.53 | 896.29 | 0.5 |

56.25 | 814.96 | 0.60259 |

F (%) | u |
---|---|

0 | 0 |

26.85 | 0.1 |

48.33 | 0.2 |

65.56 | 0.3 |

79.43 | 0.4 |

90.66 | 0.5 |

100 | 0.60259 |

Although this paper does not dwell into inner working of kinetic friction at the interface or molecular level, it attempts to quantify the effects of sliding friction on motion on a curve, a circular path considered in this paper, thus a dynamical model is developed to understand the motion of the person on the slide by deriving a differential equation which represents the motion, then the equation is solved by an analytical method to obtain the exact variation of velocity with time, the exit velocity is compared with different surface roughness and is compared with exit velocity for zero friction. The variation of normal force with the position on the slide is determined and is compared with different roughness, the critical coefficient of friction is determined and its implications are stated, however, the analysis can be extended to any other non-linear geometry like parabola or an exponential function.

Even though a detailed analysis is not shown in this paper, the model can be used to optimize the flight evacuation system which at present consists of a straight slide but instead if it were to be made like a circular then the motion of the passengers can be controlled and optimized by choosing a roughness value such that it lies within a close range of the critical coefficient of friction u_{c} such that the passengers can be made to slow down and stop exactly at the end of the slide to avoid injuries caused by tripping and falling due to loss of control.

PrahladKulkarni, (2015) A Dynamical Model to Analyze the Influence of Sliding Friction on Motion on a Curve—An Analytical Method. Open Journal of Applied Sciences,05,434-442. doi: 10.4236/ojapps.2015.58043