A 46.2-kg crate rests on a level floor at a shipping dock. The coefficients of static and kinetic friction are 0.566 and 0.440, respectively. What horizontal pushing force is required to (a) just start the crate moving and (b) slide the crate across the dock at a constant speed?
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To find the horizontal pushing force required to start the crate moving and slide it at a constant speed, we need to consider the concept of friction.
The maximum static friction is given by the equation:
\( F_{\text{static max}} = \mu_{\text{static}} \times N \)
where \( \mu_{\text{static}} \) is the coefficient of static friction and \( N \) is the normal force.
The normal force acting on the crate is equal to the weight of the crate, which can be calculated using the equation:
\( N = m \times g \)
where \( m \) is the mass of the crate and \( g \) is the acceleration due to gravity.
Given that the mass of the crate is 46.2 kg, and the coefficient of static friction is 0.566, we can calculate the maximum static frictional force:
\( F_{\text{static max}} = 0.566 \times (46.2 \times 9.8) \)
Now, let's calculate the force required to start the crate moving (a).
The force required to overcome static friction and set the crate in motion is equal to the maximum static frictional force:
\( F_{\text{start}} = F_{\text{static max}} \)
To calculate the force required to slide the crate at a constant speed (b), we need to consider the coefficient of kinetic friction.
The force of kinetic friction can be calculated using the equation:
\( F_{\text{kinetic}} = \mu_{\text{kinetic}} \times N \)
where \( \mu_{\text{kinetic}} \) is the coefficient of kinetic friction.
Given that the coefficient of kinetic friction is 0.440, we can calculate the force required to slide the crate at a constant speed:
\( F_{\text{slide}} = 0.440 \times (46.2 \times 9.8) \)
So, the answers to the questions are:
(a) The horizontal pushing force required to start the crate moving is \( F_{\text{start}} \), which is equal to \( F_{\text{static max}} \).
(b) The horizontal pushing force required to slide the crate at a constant speed is \( F_{\text{slide}} \), which is equal to \( F_{\text{kinetic}} \).