Please help me I have no idea how to do this problem at all.
Question:A sledge loaded with bricks has a total mass of 17.8 kg and is pulled at constant speed by a rope inclined at 20.8° above the horizontal. The sledge moves a distance of 20.6 m on a horizontal surface. The coefficient of kinetic friction between the sledge and surface is 0.500.
(a) What is the tension in the rope?_ N
(b) How much work is done by the rope on the sledge?__ kJ
(c) What is the mechanical energy lost due to friction?__KJ
you start by breaking up the tension in the rope into a vertical part (which reduces force normal and friction), and a horizontal part.
Forcenormal=mg-vertical component of tension
Friction=forcenormal*mu
so to move, the horizontal component of tension must at least equal fricion.
You can now find tension from that.
work on sled? tensioninhorizontal*distance.
frictionwork? friction*distance
To solve this problem, we can break it down into several steps:
Step 1: Determine the force of gravity acting on the sledge.
The force of gravity can be calculated using the equation:
F_gravity = mass * gravity
where mass is the total mass of the sledge (17.8 kg) and gravity is the acceleration due to gravity (which is approximately 9.8 m/s^2).
By substituting the values, we get:
F_gravity = 17.8 kg * 9.8 m/s^2 = 174.44 N
Step 2: Find the normal force.
The normal force is the force exerted by the horizontal surface on the sledge and is equal in magnitude but opposite in direction to the force of gravity. Since the sledge is on a horizontal surface, the normal force is equal to the force of gravity.
Therefore, the normal force is also 174.44 N.
Step 3: Calculate the force of friction.
The force of friction can be found using the equation:
F_friction = coefficient of friction * normal force
The coefficient of kinetic friction is given as 0.500 and the normal force is 174.44 N.
By substituting the values, we get:
F_friction = 0.500 * 174.44 N = 87.22 N
Step 4: Calculate the tension in the rope.
The tension in the rope can be found by resolving the forces in the vertical direction.
Tension * sin(20.8°) = F_gravity
Tension = F_gravity / sin(20.8°)
By substituting the values, we get:
Tension = 174.44 N / sin(20.8°) ≈ 510.10 N
Therefore, the tension in the rope is approximately 510.10 N.
Step 5: Calculate the work done by the rope on the sledge.
The work done by a force can be calculated using the equation:
Work = force * distance * cos(angle)
In this case, the force is equal to the tension in the rope (510.10 N), the distance is given as 20.6 m, and the angle is 0° (as the force and distance are both in the horizontal direction).
By substituting the values, we get:
Work = 510.10 N * 20.6 m * cos(0°) = 10,515.26 N·m
To convert the work to kilojoules (kJ), divide by 1000:
Work = 10,515.26 N·m / 1000 = 10.52 kJ
Therefore, the work done by the rope on the sledge is approximately 10.52 kJ.
Step 6: Calculate the mechanical energy lost due to friction.
The mechanical energy lost due to friction can be calculated using the equation:
Energy_lost = force of friction * distance
In this case, the force of friction is 87.22 N and the distance is 20.6 m.
By substituting the values, we get:
Energy_lost = 87.22 N * 20.6 m = 1,795.95 N·m
To convert the energy to kilojoules (kJ), divide by 1000:
Energy_lost = 1,795.95 N·m / 1000 = 1.80 kJ
Therefore, the mechanical energy lost due to friction is approximately 1.80 kJ.
To summarize:
(a) The tension in the rope is approximately 510.10 N.
(b) The work done by the rope on the sledge is approximately 10.52 kJ.
(c) The mechanical energy lost due to friction is approximately 1.80 kJ.
To solve this problem, we will break it down into smaller steps.
Step 1: Determine the forces acting on the sled.
In this problem, three forces are acting on the sled:
1. Tension force in the rope - This force is directed horizontally.
2. Weight force - This force acts vertically downwards and can be calculated as the mass of the sled multiplied by the acceleration due to gravity, which is approximately 9.8 m/s^2.
3. Friction force - This force acts in the opposite direction of motion and can be calculated using the coefficient of kinetic friction and the normal force, which is equal to the weight of the sled in this case.
Step 2: Resolve the forces into vertical and horizontal components.
Since we are interested in the tension in the rope, we will only consider the horizontal forces. The vertical forces will cancel each other out since the sled is moving at a constant speed.
Step 3: Use trigonometry to find the horizontal component of the weight force.
The horizontal component of the weight force can be calculated by multiplying the weight force by the sine of the angle of inclination (20.8°).
Step 4: Apply Newton's second law in the horizontal direction.
The net force in the horizontal direction is the difference between the tension force and the horizontal component of the weight force. Using Newton's second law (F = ma), where F is the net force, m is the mass of the sled, and a is the acceleration (which is zero since the sled is moving at a constant speed), we can find the tension force.
Step 5: Calculate the work done by the rope on the sled.
The work done by the tension force is given by the equation W = F * d, where W is the work done, F is the force applied, and d is the distance over which the force is applied. In this case, the distance is equal to 20.6 m.
Step 6: Calculate the mechanical energy lost due to friction.
The mechanical energy lost due to friction is equal to the work done by the friction force, which can be calculated using the equation W = f * d, where W is the work done, f is the frictional force, and d is the distance over which the force is applied.
Now, let's apply these steps to solve the problem:
Step 1:
Weight force = mass * acceleration due to gravity = 17.8 kg * 9.8 m/s^2
Friction force = coefficient of friction * normal force
Normal force = weight force = 17.8 kg * 9.8 m/s^2
Step 2:
Horizontal component of the weight force = weight force * sin(angle of inclination)
Step 3:
Net force in the horizontal direction = tension force - horizontal component of the weight force
Step 4:
Using Newton's second law: Net force = mass * acceleration
Rearranging the equation: Tension force = mass * acceleration + horizontal component of the weight force
Step 5:
Work done by the rope on the sled = tension force * distance
Step 6:
Mechanical energy lost due to friction = friction force * distance
Plug in the values you calculated in each of the steps to find the answers to parts (a), (b), and (c) of the question.