A factory worker pushes a 29.2kg crate a distance of 4.1m along a level floor at constant velocity by pushing downward at an angle of 28 degrees below the horizontal. The coefficient of kinetic friction between the crate and floor is 0.25.
(a) What magnitude of force must the worker apply?
(b) How much work is done on the crate by this force?
(c) How much work is done on the crate by frication?
(d) How much work is done on the crate by normal force?
(e) How much work is done on the crate by the gravity?
(f) What is the total work done on the crate?
weight = m g
normal force down on floor N= m g+Fsin 28
friction force back = .25 N
so
F cos 28 = friction force back
(a) F = friction force back / cos 28
force component in direction of motion = F cos 28 Newtons
d = 4.1 meters
so
(b) Win = 4.1 F cos 28 Joules
(c) -4.1 F cos 28
(d) no vertical motion so = ZERO
(e) same as (d) ZERO
(f) zero, there is no net force on the crate so the work in from the push is equal and opposite to that out by friction.
To solve this problem, we can break it down into smaller steps. Let's go step by step:
Step 1: Find the horizontal and vertical components of the worker's applied force.
The horizontal component of the applied force can be found using the angle of 28 degrees below the horizontal. We can use trigonometry to find it:
Horizontal component = Force * cos(angle)
Horizontal component = Force * cos(28)
Similarly, the vertical component of the force can be found:
Vertical component = Force * sin(angle)
Vertical component = Force * sin(28)
Step 2: Determine the force of kinetic friction.
The force of kinetic friction can be found using the formula:
Force of friction = coefficient of friction * Normal force
Given that the coefficient of kinetic friction is 0.25, and the normal force is equal to the weight of the crate (mg), we can calculate the force of friction as follows:
Force of friction = 0.25 * (mass of the crate * acceleration due to gravity)
Force of friction = 0.25 * (29.2 kg * 9.8 m/s^2)
Step 3: Calculate the work done by each force.
(a) The magnitude of force applied by the worker can be calculated as the sum of the horizontal and vertical components found in Step 1.
(b) The work done on the crate by the applied force can be found using the formula:
Work = Magnitude of force * distance * cos(angle)
(c) The work done on the crate by friction can be calculated using the formula:
Work = Force of friction * distance * cos(angle)
(d) The work done on the crate by the normal force is zero because the normal force acts perpendicular to the displacement.
(e) The work done on the crate by gravity can be calculated using the formula:
Work = (mass of the crate * acceleration due to gravity) * distance * cos(angle)
(f) The total work done on the crate is the sum of the work done by all forces.
Let's solve each of these steps one by one:
Step 1:
Horizontal component = Force * cos(28) = F * cos(28)
Vertical component = Force * sin(28) = F * sin(28)
Step 2:
Force of friction = 0.25 * (29.2 kg * 9.8 m/s^2) = 0.25 * (285.76 N) = 71.44 N
Step 3:
(a) Magnitude of force applied = Horizontal component + Vertical component = F * cos(28) + F * sin(28) = F * (cos(28) + sin(28))
(b) Work done on the crate by the applied force = Magnitude of force applied * distance * cos(28) = (F * (cos(28) + sin(28))) * 4.1 m * cos(28)
(c) Work done on the crate by friction = Force of friction * distance * cos(28) = 71.44 N * 4.1 m * cos(28)
(d) Work done on the crate by the normal force = 0 J (as explained earlier)
(e) Work done on the crate by gravity = (mass of the crate * acceleration due to gravity) * distance * cos(28) = (29.2 kg * 9.8 m/s^2) * 4.1 m * cos(28)
(f) Total work done on the crate = Work done by the applied force + Work done by friction + Work done by gravity
This should give you the step-by-step solution to the problem.
To solve this problem, we will use Newton's laws of motion and the work-energy principle.
(a) To find the magnitude of force that the worker must apply, we need to calculate the net force acting on the crate. At constant velocity, the net force is zero. The forces acting on the crate are the force applied by the worker and the force due to friction.
The force of friction can be calculated using the formula:
Frictional force (F_friction) = coefficient of friction (μ) * normal force (N)
In this case, the normal force is equal in magnitude to the weight of the crate, which can be calculated as:
Weight (W) = mass (m) * gravitational acceleration (g)
Given:
Mass of the crate (m) = 29.2 kg
Gravitational acceleration (g) = 9.8 m/s^2
Now, let's calculate the normal force:
Normal force (N) = Weight (W) = m * g = 29.2 kg * 9.8 m/s^2
Next, we can calculate the frictional force:
F_friction = μ * N = 0.25 * (29.2 kg * 9.8 m/s^2)
To find the magnitude of force applied by the worker, we need to equate the frictional force with the force applied by the worker:
Force applied by the worker = F_friction
(b) The work done on the crate by the worker's force can be calculated using the formula:
Work = Force * distance * cos(angle)
Given the angle at which the worker is pushing the crate (θ = 28 degrees) and the distance (d = 4.1 m), we can use:
Work = F * d * cos(θ)
Substitute the value of the force obtained in part (a) to calculate the work done by the worker.
(c) The work done on the crate by friction can be calculated using the formula:
Work = Force * distance * cos(180 degrees)
Since the frictional force is acting opposite to the direction of displacement, the angle between the force and displacement is 180 degrees. Substitute the value of the frictional force obtained in part (a) to calculate the work done by friction.
(d) The work done on the crate by the normal force is zero. The normal force acts perpendicular to the displacement, and thus, there is no component of force in the direction of displacement.
(e) The work done on the crate by gravity can be calculated using the formula:
Work = Weight * distance * cos(angle)
Given the angle (θ = 0 degrees) and the distance (d = 4.1 m), we can use:
Work = W * d * cos(θ)
Substitute the value of weight obtained earlier to calculate the work done by gravity.
(f) The total work done on the crate is the sum of the work done by the worker, the work done by friction, and the work done by gravity.
Total work = Work by worker + Work by friction + Work by gravity
Now, let's plug in the values and calculate each part of the problem.