Forces Examples
Example 1.
Two cows pull on a cart. One cow pulls to the east with a force of 3000 N. The other cow pulls to the north with 4000 N. Find the resultant force on the cart.
Answer.
Since force is a vector with magnitude and direction, we must find both of these.
Use arrows to represent the forces. Draw an arrow to the right to show the force to the east, and label it '3000 N'. Draw another arrow upwards for the force to the north, label it '4000 N'. It helps if you make this second arrow a bit longer, to make it obvious that it is the larger force. Draw the two arrows so that their tail ends touch each other. Since east and north are at 90 degrees, write this down also. We now have all the information on a picture. This is always convenient.
Take the two arrows as the sides of a parallelogram. Draw the opposite sides to complete this parallelogram. In this case, we just get a rectangle.
Draw an arrow from the tail ends of the first two arrows, to the opposite corner of the rectangle. This is just the diagonal. The length and direction of this arrow will help us to find the resultant force.
The diagonal arrow divides the rectangle into two triangles. Look at the triangle OAC. This is a right angled triangle. Think of the magnitude of the two original forces as lengths. So the length of OA is 3000 N, and the length of AC is 4000 N. Since OAC is a right angled triangle, we can use Pythagoras theorem to find the length for the resultant force OC. Let this be F N.
So F N = 5000 N. This is the magnitude of the resultant force. Next, we need to find its direction.
There could be more than one way to specify the direction. We can give the answer as the angle that the resultant force makes with east, or the angle that it makes with the north. Or anything else you can think of. It is fine as long as it is clear.
So lets find the angle COA, the angle that the resultant force makes with the east. Let this angle be θ. We need to use trigonometry now. Since OAC is a right angled triangle,
Using a calculator, we find θ = 53.13°.
So the answer is: the resultant force of the two cows has
a magnitude of 5000 N, and a direction that is 53.13° to the east.
Example 2.
Two cows pull on a cart. One cow pulls to the east, and the other cow pulls to the north. The resultant force of the two cows has a magnitude of 5000 N, and a direction that is 53.13° to the east. Find the force of each of the cows.
Answer.
We need to resolve the resultant force into two components, one to the east and one to the north. This just means that if we combine the two component forces, we should get back the original force as a resultant.
Draw an arrow to represent the 5000 N force. Draw a dashed line horizontally, starting from the tail end of the arrow. This is to show the east direction. Draw a dashed line vertically, also from the tail end of the arrow, for the north direction. Label the arrow '5000 N'. Label the angle with the horizontal line '53.13°'. It really helps to have all the information on the picture.
From the tip of the arrow, drop a perpendicular to the horizontal line. Draw a new arrow from the tail of the original arrow, to the foot of the perpendicular. This represents the force from the cow that pulls to the east. Let this force be x.
Go back to the tip of the original arrow. Drop a perpendicular to the vertical line. Draw another new arrow from the tail of the original arrow, to the foot of this perpendicular. This represents the force from the cow that pulls to the north. Let this force be y.
We now have a rectangle. Think of the '5000 N' as the length of the resultant force. We must now find the length of x and y. The resultant force divides the rectangle into two triangles. Look at the lower one - triangle OAC. This is a right angled triangle. We know the longest side and one of the acute angles. So we can find the other two sides using trigonometry.
Using the calculator, we will find that x is 3000.0 N and y is 4000.0 N.
So one cow pulls to the east with 3000.0 N, the other cow
pulls to the north with 4000.0 N.
Example 3.
Two cows pull on a cart. One cow pulls to the east with a force of 3000 N. The other cow pulls to the north with 4000 N. A third cow now pulls at the cart with a force that balances the first two, so that the cart does not move. Find the force of the third cow.
Answer.
The three forces are in equilibrium. Draw a small circle to represent the cart. Draw the force from the first cow as an arrow from the cart, pointing to the right. Label this arrow A. Draw the force from the second cow as an arrow from the cart, pointing upwards. Label this arrow B. Draw the force from the third cow as an arrow pointing roughly in the opposite direction to the first two arrows. Label this arrow F.
Now forget about the cart. Imagine that the arrows are real objects. You are allowed to move them around, but you must not change their directions. We need to move them until the head of each arrow touches the tail of another arrow. They can be in any order. Lets start by moving arrow A up until its tail touches the head of arrow B. Next, shift arrow F until its tail touches the head of arrow A. Since the forces are all balanced, that is in equilibrium, they must form a closed triangle. This means that the head of arrow F must touch the tail off arrow B.
The arrows now form a triangle. Notice that arrows A and B are at right angles, because one points to the east and the other points to the north. So we have a right angled triangle. We know that A is 3000 N and B is 4000 N. Then we can use Pythagoras theorem to find F:
This gives F = 5000 N. We also need to find its direction. Lets use the angle θ between F and A. Since A points from west to east, θ is the angle between F and the west. Using trigonometry,
Using a calculator, we find θ = 53.13°.
So the third cow must apply a force of 5000 N,
in a direction of 53.13° to the west.
Copyright 2011 by Kai Hock. All rights reserved.
Last updated: 10 July 2011