A 65 kg crate is dragged across a floor by pulling on a rope attached to the crate and inclined 25° above the horizontal. (a) If the coefficient of static friction is 0.54, what minimum force magnitude is required from the rope to start the crate moving? (b) If ìk = 0.36, what is the magnitude of the initial acceleration (m/s^2) of the crate?
(a) Fmin cos25 = M g*(1 - sin25)*0.54
(b) Using Fmin from part (a), solve for a:
Fnet = Fmin - M g*(1 - sin25)*0.36
= M a
27
To find the answers to these questions, we can use the concept of forces and Newton's laws of motion. First, let's break down the problem and analyze the forces acting on the crate.
(a) To find the minimum force required to start the crate moving, we need to consider the force of static friction. The force of static friction can be calculated using the formula:
fs = μs * N
where fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force.
The normal force is the force perpendicular to the surface, which in this case is equal to the weight of the crate. The weight of the crate can be calculated as:
N = m * g
where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).
In this case, the crate is inclined at an angle of 25° above the horizontal. Therefore, the normal force N can be calculated as:
N = m * g * cos(θ)
where θ is the angle of inclination.
Now we can substitute the values into the equation for the force of static friction:
fs = μs * N
fs = μs * m * g * cos(θ)
Finally, the minimum force required to start the crate moving is equal to the force of static friction:
F = fs
F = μs * m * g * cos(θ)
Substituting the given values, we can solve for the minimum force magnitude.
(b) Once the crate starts moving, the force of kinetic friction comes into play. The force of kinetic friction can be calculated using the formula:
fk = μk * N
where fk is the force of kinetic friction and μk is the coefficient of kinetic friction.
Similar to the previous case, the normal force N can be calculated as:
N = m * g * cos(θ)
Now we can substitute the values into the equation for the force of kinetic friction:
fk = μk * N
fk = μk * m * g * cos(θ)
The acceleration of the crate can be calculated using Newton's second law of motion:
F - fk = m * a
where F is the force applied to the crate and a is the acceleration.
Rearranging the equation, we can solve for the initial acceleration:
a = (F - fk) / m
Substituting the given values, we can solve for the magnitude of the initial acceleration of the crate.
Note: Make sure to convert all angles to radians when using trigonometric functions in calculations.