Guide 5-2.  Solving a Net Force Problem in Two Dimensions

The following example problem illustrates the method for solving a net force problem in two dimensions. The only difference from a 1-dimensional problem is that you have to write and solve net force equations for each axis.

Problem:  A child pulls a wagon with a force of 44 N by a handle making an angle of 29o with the horizontal. If the wagon has a mass of 4.5 kg, what is the acceleration of the wagon?  What is the force with which the ground pushes up on the wagon?

We begin by drawing a picture (to the right) with the directions of velocity and acceleration shown. Then we draw a force diagram and set up x- and y-axes as shown below.  Then we write the given and the goal.

Forces on the wagon

Given

T = 44 N  (Note that the pull force is a tension force.)
m = 4.5 kg
θ = 29o

Goal:

Find the acceleration, a, of the wagon
Find the normal force, N, of the ground on the wagon

Solution:

Now we write net force equations.

Fnet,x = Tcosθ

Fnet,y = Tsinθ + N - mg

Using Newton's 2nd Law in the x-direction,

max = Tcosθ,

and solving for ax,

ax = Tcosθ/m.

The acceleration is ax = (44 N)cos29o/4.5 kg = 8.6 m/s2.  That's a pretty amazing acceleration for a child.

Using Newton's 2nd Law in the y-direction,

may = Tsinθ + N - mg.

The vertical acceleration is 0, so

0 = Tsinθ + N - mg,

and solving for N,

N = mg - Tsinθ

The normal force is N = (4.5 kg)(9.8 m/s2) - (44 N)sin29o = 23 N.

Checks:  The units work out since N = kgm/s2.  Acceleration is positive which is to the right, as expected.  The normal force is positive, since we're treating it as a magnitude.  The value of the normal force is less than the weight, since the vertical component of the tension force works together with the normal force to balance the weight of the wagon.