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 ![]() ![]() ![]() |
![]() |
gravitational force | unconditional | ![]() |
vertical, downward |
![]() |
normal force | assuming that ![]() |
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 ![]() ![]() |
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: