crate weighing 6.90 x 10^3 N is pulled up a 46° incline by a force parallel to the plane. If the coefficient of kinetic friction between the crate and the surface of the plane is 0.750, what is the magnitude of the applied force required to move the crate at a constant velocity?
Assuming the crate moves upwards.
The applied force P is upwards whether the crate moves upwards or downwards at a constant velocity.
coeff. of kinetic friction, μ = 0.75
angle of incline, θ = 46° (with horizontal)
If crate moves upwards with a constant velocity, then forces along plane sums to zero (no net force).
The resistive (friction) force acts down the plane in opposite direction to the motion.
Resolve weight normal to plane,
N=W*cos(θ)
Force down the plane due to weight (downwards)
F=W*sin(θ)
Frictional force (downwards)
Fr=μN = μW cos(θ)
Applied force, P (upwards)
Sum forces along plane:
P-Fr-F=0
P=μW cos(θ)+W sin(θ)
Substitute values and evaluate P.
I get about 8558N.
To find the magnitude of the applied force required to move the crate at a constant velocity, we need to consider the forces acting on the crate.
Let's break down the problem step by step:
Step 1: Draw a free-body diagram
The first step is to draw a free-body diagram, which shows all the forces acting on the crate. In this case, we have the following forces:
- The weight of the crate acting vertically downward (6.90 x 10^3 N).
- The normal force acting perpendicular to the incline.
- The force of kinetic friction acting parallel to the incline.
- The force applied parallel to the incline.
Step 2: Resolve the weight force
Since the force is acting on an incline, we need to resolve the weight force into two components: one perpendicular to the incline (normal force) and one parallel to the incline (downhill force). The downhill force can be calculated as the weight of the crate multiplied by the sine of the angle of the incline (46 degrees in this case).
Downhill force = Weight of crate * sin(angle of incline)
Downhill force = 6.90 x 10^3 N * sin(46°)
Step 3: Calculate the force of kinetic friction
The force of kinetic friction can be calculated as the coefficient of kinetic friction multiplied by the normal force. The normal force can be calculated as the weight of the crate multiplied by the cosine of the angle of the incline.
Normal force = Weight of crate * cos(angle of incline)
Normal force = 6.90 x 10^3 N * cos(46°)
Force of kinetic friction = Coefficient of kinetic friction * Normal force
Force of kinetic friction = 0.750 * [6.90 x 10^3 N * cos(46°)]
Step 4: Equilibrium condition
For an object to move at a constant velocity, the net force acting on it must be zero. In this case, the applied force and the force of kinetic friction are in opposite directions, so we can write the equation:
Applied force - Force of kinetic friction = 0
Step 5: Solve for the applied force
Rearranging the equation, we can calculate the magnitude of the applied force:
Applied force = Force of kinetic friction
Substituting the values we calculated, we get:
Applied force = 0.750 * [6.90 x 10^3 N * cos(46°)]
Now you can substitute the numbers into the equation and calculate the magnitude of the applied force.