# Dragging problem

This article discusses a scenario/arrangement whose statics/dynamics/kinematics can be understood using the ideas of classical mechanics.

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## Contents

## The scenario

Suppose a block rests on a fixed horizontal floor. The limiting coefficient of static friction between and the floor is and the coefficient of kinetic friction is . A force is applied on the block in a diagonal direction at an angle with the horizontal. The questions are as follows:

- For a given angle , what is the minimum magnitude of force needed to get the block to start sliding, and how is the acceleration of the block given as a function of ?
- For what value of is the minimum magnitude of force needed to get the block to start sliding as low as possible, and what is this minimum magnitude of force?
- For what value of is the minimum magnitude of force needed to get the block to
*keep*sliding as low as possible, and what is this minimum magnitude of force?

## Basic components of force diagram

Force (letter) | Nature of force | Condition for existence | Magnitude | Direction |
---|---|---|---|---|

external force being applied | given to us that it's being applied | angle with horizontal. The horizontal component is and the vertical component is . | ||

gravitational force | unconditional | vertical, downward | ||

normal force | assuming that , otherwise the block will lift off the floor. | adjusts so that there is no net force in the vertical direction | vertical, upward | |

static friction | no sliding | adjusts so that there is no net force in the horizontal direction | horizontal, opposite to horizontal component of | |

kinetic friction | sliding | horizontal, opposite to direction of slipping. Assuming initially at rest, direction of slipping = horizontal component of , so this is opposite to horizontal component of . |

### Vertical components

Taking components in the vertical direction, and assuming no acceleration in the vertical direction (because the block does not lift off), we get:

### Horizontal components and solution assuming no sliding

If there is no sliding, we get:

We also have:

Plugging in from (1.1) into (2.2) and plugging from this into (2.1) gives:

This simplifies to:

This is a necessary and sufficient condition for there to be no sliding *assuming* the block was initially at rest.

### Horizontal components and solution assuming sliding

If there is sliding along the horizontal component of the direction of , denote by the acceleration along the horizontal component of the direction of . We get:

We also have:

Plugging in from (1.1) into (3.2) and plugging from this into (3.1) gives:

This simplifies to:

## Angle needed to minimize the minimum force needed to start sliding

As noted in equation (2.4), the minimum force needed to *start* sliding is:

The numerator is constant, so to minimize this is equvalent to maximizing the denominator, for . In other words, we need to maximize:

Using either differential calculus or basic trigonometry or the Cauchy-Schwarz inequality, we get that the maximum occurs when , and the value of the maximum is , so that the minimum possible value of the minimum force needed is: