This part of Lesson 3 focuses on net force-acceleration
problems in which an applied force is directed at an angle
to the horizontal. We have already discussed earlier
in Lesson 3 how a force directed an angle can be
resolved into two components - a horizontal and a vertical
component. We have also discussed in an earlier unit that
the acceleration of an object is related to the net force
acting upon the object and the mass of the object (Newton's
second law). We had used this principle to solve
net force-acceleration
problems in an earlier unit. Therefore, it is a natural
extension of this unit to combine our understanding of
Newton's second law with our understanding of force vectors
directed at angles.

Consider the situation below in which a
force is directed at an angle to the horizontal. In such a
situation, the applied force can be resolved into two
components. These two components can be considered to
replace the applied force at an angle. By doing so, the
situation simplifies into a familiar situation in which all
the forces are directed horizontally and vertically.

Once the situation has been simplified,
the problem can be solved like any other problem. The task
of determining the acceleration involves first determining
the net force by adding up all the forces as vectors and
then dividing the net force by the mass to determine the
acceleration. In the above situation, the vertical forces
are balanced (i.e., F_{grav}, F_{y}, and
F_{norm} add up to 0 N), and the horizontal forces
add up to 29.3 N, right (i.e., 69.3 N, right + 40 N, left =
29.3 N, right). The net force is 29.3 N, right and the mass
is 10 kg (m = F_{grav}/g); therefore, the
acceleration is 2.93 m/s/s, right.

To test your understanding, analyze the
two situations below to determine the net force and the
acceleration. When finished, click the button to view the
answers.

There is one peculiarity about these types
of problems which you need to be aware of. The normal
force (F_{norm})
is not necessarily equal to the gravitational
force (F_{grav})
as it has been in problems which we have previously seen.
The principle is that the vertical forces must balance if
there is no vertical acceleration. If an object is being
dragged across a horizontal surface, then there is no
vertical acceleration. For this reason, the normal force
(F_{norm}) plus the vertical component
(F_{y}) of the applied force must balance the
gravitational force (F_{grav}). A quick review of
these problems shows that this is the case. If there is an
acceleration for an object being pulled across a floor, then
it is a horizontal acceleration; and thus the only
imbalance of force would be in the horizontal
direction.

Now consider the following situation in
which a force analysis must be conducted to fill in all the
blanks and to determine the net force and acceleration. In a
case such as this, a thorough understanding of the
relationships between the various quantities must be fully
understood. Make an effort to solve this problem. When
finished, click the button to view the answers. (When you
run into difficulties, consult the help
from a previous unit.)

In conclusion, a situation involving a force at an angle
can be simplified by using trigonometric relations to
resolve that force into two components. Such a situation can
be analyzed like any other situation involving individual
forces. The net force can be determined by adding all the
forces as vectors and the acceleration can be determined as
the ratio of Fnet/mass.

Check
Your Understanding

The following problems provide plenty of practice with
F_{net}= m • a problems involving forces at
angles. Try each problem and then click the button to view
the answers.

1.
A 50-N applied force (30 degrees to the horizontal)
accelerates a box across a horizontal sheet of ice (see
diagram). Glen Brook, Olive N. Glenveau, and Warren Peace
are discussing the problem. Glen suggests that the normal
force is 50 N; Olive suggests that the normal force in the
diagram is 75 N; and Warren suggests that the normal force
is 100 N. While all three answers may seem reasonable, only
one is correct. Indicate which two answers are wrong and
explain why they are wrong.

2. A box is pulled at a constant speed of 0.40 m/s across
a frictional surface. Perform an extensive analysis of the
diagram below to determine the values for the blanks.

3. Use your understanding of force relationships and
vector components to fill in the blanks in the following
diagram and to determine the net force and
acceleration of the object. (F_{net} = m*a;
F_{frict} = "mu"*F_{norm}; F_{grav}
= m*g)

4. The 5-kg mass below is moving with a constant speed of
4 m/s to the right. Use your understanding of force
relationships and vector components to fill in the blanks in
the following diagram and to determine the net force
and acceleration of the object. (F_{net} = m*a;
F_{frict} = "mu"*F_{norm}; F_{grav}
= m*g)

5. The following object is being pulled at a constant
speed of 2.5 m/s. Use your understanding of force
relationships and vector components to fill in the blanks in
the following diagram and to determine the net force
and acceleration of the object. (F_{net} = m*a;
F_{frict} = "mu"*F_{norm}; F_{grav}
= m*g)

6. Use your understanding of force relationships and
vector components to fill in the blanks in the following
diagram and to determine the net force and
acceleration of the object. (F_{net} = m*a;
F_{frict} = "mu"*F_{norm}; F_{grav}
= m*g)

7. Study the diagram below and determine the acceleration
of the box and its velocity after being pulled by the
applied force for 2.0 seconds.

8. A student pulls a 2-kg backpack across the ice (assume
friction-free) by pulling at a 30-degree angle to the
horizontal. The velocity-time graph for the motion is shown.
Perform a careful analysis of the situation and determine
the applied force.

9. The following object is moving to the right and
encountering the following forces. Use your understanding of
force relationships and vector components to fill in the
blanks in the following diagram and to determine the
net force and acceleration of the object. (F_{net} =
m*a; F_{frict} = "mu"*F_{norm};
F_{grav} = m*g)

10. The 10-kg object is being pulled to the left at a
constant speed of 2.5 m/s. Use your understanding of force
relationships and vector components to fill in the blanks in
the following diagram. (F_{net} = m*a;
F_{frict} = "mu"*F_{norm}; F_{grav}
= m*g)

11. Use your understanding of force relationships and
vector components to fill in the blanks in the following
diagram and to determine the net force and
acceleration of the object. (F_{net} = m*a;
F_{frict} = "mu"*F_{norm}; F_{grav}
= m*g)