A 50.5-kg crate rests on a level floor at a shipping dock. The coefficients of static and kinetic friction are 0.624 and 0.364, 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?
Wb = M*g = 50.5 * 9.8 = 494.9 N. = Normal force, Fn.
Fs = u*Fn = 0.624 * 494.9 = 308.8 N. =
Force of static friction.
Fk = 0.364 * 494.9 = 180.1 N.
a. F-Fs = M*a.
F-308.8 = M*0 = 0
F = 308.8 N.
b. F-Fk = M*a.
F-180.1 = M*0 = 0.
F = 180.1 N.
To calculate the required horizontal pushing force to (a) just start the crate moving and (b) slide the crate across the dock at a constant speed, we need to consider the forces acting on the crate.
Let's denote the coefficient of static friction as μs and the coefficient of kinetic friction as μk.
(a) Just starting the crate moving:
To determine the force required to just start the crate moving, we need to calculate the maximum static friction force (fs_max). The formula for the maximum static friction force is given by:
fs_max = μs * Fn
Where:
- fs_max is the maximum static friction force,
- μs is the coefficient of static friction, and
- Fn is the normal force on the crate.
1. Calculate the normal force:
The normal force (Fn) is equal to the weight of the crate, which can be calculated as:
Fn = m * g
Where:
- m is the mass of the crate (50.5 kg), and
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fn = 50.5 kg * 9.8 m/s^2
Fn = 495.9 N
2. Calculate the maximum static friction force:
Using the formula mentioned above, we can calculate the maximum static friction force:
fs_max = μs * Fn
fs_max = 0.624 * 495.9 N
Now we have the maximum static friction force required to get the crate moving.
(b) Sliding the crate across the dock at a constant speed:
To determine the force required to slide the crate at a constant speed, we need to calculate the kinetic friction force (fk). The kinetic friction force is given by:
fk = μk * Fn
Where:
- fk is the kinetic friction force,
- μk is the coefficient of kinetic friction, and
- Fn is the normal force on the crate.
1. Calculate the kinetic friction force:
Using the formula mentioned above, we can calculate the kinetic friction force:
fk = μk * Fn
fk = 0.364 * 495.9 N
Now we have the kinetic friction force required to slide the crate at a constant speed.
Remember that the pushing force required to overcome static friction and start the crate moving will be greater than the force required to keep the crate sliding at a constant speed due to the lower coefficient of kinetic friction.
To determine the horizontal pushing force required to start and slide the crate, we need to analyze the forces acting on it. Let's break it down step by step:
a) To just start the crate moving, we need to overcome the static friction. The formula to calculate the static friction force is:
F_static = μ_static * N
where F_static is the static friction force, μ_static is the coefficient of static friction, and N is the normal force.
The normal force (N) is equal to the weight of the crate since the crate rests on a level floor with no vertical acceleration. The weight can be calculated using the formula:
weight = mass * gravity
where mass is the mass of the crate (given as 50.5 kg) and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, let's calculate the force required to start the crate moving:
F_static = μ_static * N
F_static = 0.624 * (mass * gravity)
F_static = 0.624 * (50.5 kg * 9.8 m/s^2)
b) To slide the crate across the dock at a constant speed, we need to overcome the kinetic friction. The formula to calculate the kinetic friction force is similar to the static friction force:
F_kinetic = μ_kinetic * N
where F_kinetic is the kinetic friction force, μ_kinetic is the coefficient of kinetic friction, and N is still the normal force.
Now, let's calculate the force required to slide the crate:
F_kinetic = μ_kinetic * N
F_kinetic = 0.364 * (mass * gravity)
F_kinetic = 0.364 * (50.5 kg * 9.8 m/s^2)
By plugging in the given values and performing the calculations, we can find the respective forces required to start and slide the crate across the dock at a constant speed.
Alright, let's take a look at this crate situation. Hold on to your packing peanuts!
(a) To get this crate moving, we need to overcome the static friction. The formula to calculate the maximum static friction force is given by:
Fs = μs * Fn
Where Fs is the static friction force, μs is the coefficient of static friction, and Fn is the normal force (equal to the weight of the crate).
Given that the coefficient of static friction (μs) is 0.624 and the weight of the crate (Fn) is:
Fn = mass * gravity
Fn = 50.5 kg * 9.8 m/s^2
Now we can substitute values:
Fs = 0.624 * (50.5 kg * 9.8 m/s^2)
Calculate that out and we find:
Fs ≈ 307.96 N
That's quite a push!
(b) Once the crate is in motion, we need to consider the kinetic friction force, which is slightly less than the static friction force. The formula for kinetic friction force is given by:
Fk = μk * Fn
Where Fk is the kinetic friction force, μk is the coefficient of kinetic friction (which is 0.364 in our case), and Fn is still the normal force.
Substituting the values:
Fk = 0.364 * (50.5 kg * 9.8 m/s^2)
Calculating that out, we find:
Fk ≈ 177.77 N
So to slide that crate at a constant speed, you'll need to apply a force of approximately 177.77 Newtons.
Remember to handle with care, and always keep those clown shoes on for extra traction!