To find the force the boy must exert at an angle of 35 degrees with respect to the hill, we need to break down the forces acting on the sled and use trigonometry.
First, let's identify the forces involved:
1. The force of gravity (weight) acting vertically downwards = 100 N.
2. The normal force acting perpendicular to the hill.
3. The force of friction acting parallel to the hill.
4. The force the boy exerts at an angle of 35 degrees.
Since the sled is being dragged up the slope at a constant velocity, we know that the net force acting on the sled is zero. Therefore, the force the boy exerts should balance out the forces of friction and gravity.
Let's break down the forces into their components:
1. The force of gravity can be divided into two components:
- The component acting parallel to the hill = mg * sin(20°)
- The component acting perpendicular to the hill = mg * cos(20°)
2. The force the boy exerts can be divided into two components:
- The component acting parallel to the hill = F * cos(35°)
- The component acting perpendicular to the hill = F * sin(35°)
Now, let's write the equation for the net force:
Net force = Force uphill - Force downhill
Since the sled is at a constant velocity, the forces uphill and downhill will be equal in magnitude but opposite in direction.
Force uphill = force due to the component parallel to the hill - force due to friction
= F * cos(35°) - (Coefficient of friction * normal force)
Force downhill = force due to the component parallel to the hill + force due to gravity parallel to the hill
= F * cos(35°) + mg * sin(20°)
Since the net force is zero, we can set Force uphill equal to Force downhill and solve for F:
F * cos(35°) - (Coefficient of friction * normal force) = F * cos(35°) + mg * sin(20°)
We can replace the normal force with mg * cos(20°):
F * cos(35°) - (Coefficient of friction * mg * cos(20°)) = F * cos(35°) + mg * sin(20°)
Now, let's solve this equation for F:
F * cos(35°) - F * cos(35°) = mg * sin(20°) + (Coefficient of friction * mg * cos(20°))
0 = mg * sin(20°) + (Coefficient of friction * mg * cos(20°))
Now, we can solve for F:
F = [mg * sin(20°)] / [cos(35°) - Coefficient of friction * cos(20°)]
By plugging in the given values, we can calculate the force the boy must exert at an angle of 35 degrees with respect to the hill.