1. LESSON 23 Forces Applied at Angles
Sometimes, we apply a force to an object at an angle. Imagine pushing a lawnmower or pulling a sled. In these scenarios, the direction of the applied force is at an angle to the direction of the motion of the object. In this case, we will need to resolve the angled vector and determine the x and y components (you have already learned this to an extent).
First, consider an applied force vector that is 20° above the horizon. You will need to treat the vector as the hypotenuse of a right angled triangle. You will then solve for the x and y components using trigonometry.
Fapp
20°

2. When you figure out the x and y components of a vector at an angle, you are RESOLVING the vector.
You could also be given the angle from the vertical. For instance, the last vector is 20° from the horizon, which means that it is also 70° from the vertical. Resolve the same vector using the 70° angle.
You can substitute any value for Fapp and the x and y values will remain the same (since the vector is the same). The only thing that changed between the two diagrams is the angle you were given.
Remember that the vector adjacent to the angle is Fapp∙cosθ, while the vector opposite to the angle is Fapp∙sinθ.
Fapp
70°

3. Pay attention to the direction of the applied force vector
Pay attention to the direction of the applied force vector. The x and y resolved vectors must point in the same direction.
PRACTICE: Resolve the following vector, i.e. determine the x and y values.
35°
Fapp

4. There is no standard equation that will work for these types of force problems. You will have to consider all of the x and y components in order to determine the net force and therefore, the acceleration of the object.
Suppose you pushed a lawnmower with a certain force. The handle would be at an angle to the ground, so the force would get “diluted.”
The up/down forces (y components) would consist of the normal force (upwards) and the force of gravity added to the diluted applied force (downwards). Since an object such as a lawnmower does not enter or leave the ground, the upwards and downwards forces are equal in magnitude and opposite in direction.
The left/right forces (x components) would consist of the diluted applied force minus the force of friction. This results in a net force, from which we can determine the acceleration of the object (if we know its mass).
The trick to solving these problems is to draw a proper free body diagram. The first one you draw will contain a lot of information and may get confusing. You should then summarize all of the information in a second vector diagram. This will make the problem easier to solve.

5. PROBLEM: A lawnmower is pushed with a force of 56
PROBLEM: A lawnmower is pushed with a force of 56.9 N at an angle of 31° above the ground. If the lawnmower experiences a force of friction of 24.1 N, what is the net force in the forward direction? Draw a free body diagram.

6. PROBLEM: What is the acceleration of the lawnmower in the previous question if it has a mass of 17.6 kg?
PROBLEM: What is the value of the normal force of the lawnmower? Remember that Fg = m∙g.

7. PROBLEM: A dog pulls a 12.9 kg sled at an angle of 15° above the ground with a force of 34.1 N. If the sled experiences a force of friction of 11.0 N, what is its acceleration? Draw a free body diagram.

8. PROBLEM: What is the value of the normal force in the previous problem
PROBLEM: What is the value of the normal force in the previous problem? Remember that Fg = m∙g.

9. PRACTICE: A 47. 2 kg wheelbarrow is pushed with a force of 72
PRACTICE: A 47.2 kg wheelbarrow is pushed with a force of 72.4 N at an angle of 38° from the vertical. If the wheelbarrow experiences a force of friction of 23.9 N, what is the acceleration of the wheelbarrow? Draw a free body diagram.

10. PROBLEM: What is the value of the normal force in the previous problem
PROBLEM: What is the value of the normal force in the previous problem? Remember that Fg = m∙g.

11. H.W. – Practice using the following questions.
A 41.1 kg lawnmower is pushed with a force of 72.3 N at an angle of 23° above the ground. If the lawnmower experiences a force of friction of 21.6 N, what are the values for net force, acceleration, and the normal force? Draw a free body diagram.
A child pulls a 6.3 kg wagon with a force of 31.2 N at an angle of 19° above the ground. If the wagon experiences a force of friction of 10.1 N, what are the values for the net force, the acceleration, and the normal force? Draw a free body diagram.
A 21.4 kg moving box is pushed with a force of 54.1 N at an angle of 37° from the vertical. If the box experiences a force of friction of 23.4 N, what are the values for the net force, the acceleration, and the normal force? Draw a free body diagram.
ANSWERS: 1) FNET = 45 N, a = 1.1 m/s2, FN = 430 N
2) FNET = 19 N, a = 3.0 m/s2, FN = 52 N
3) FNET = 9.2 N, a = 0.43 m/s2, FN = 250 N