# Mechanics

calculus-based physics for scientists and engineers

## Newton's Second Law in Two Dimensions

### Newton's Second Law in Two Dimensions: Example

There isn't any new physics looking at Newton's Second Law in more dimensions. All we have to work with is our working definition:

The vector sum of the forces on one object is equal to the mass of that one object times the acceleration of that one object.

We should put it to good use. The approach used in one dimension is still appropriate.

1. Determine a single object.
2. Find all forces on that one object.
3. Find vector sum of all the forces (use free-body diagram).
4. Set equal to the mass of that one object times the acceleration of that one object (also a vector).
5. Extract scalar relationships that can connect what you know with what you want to find.
Using vectors in two dimensions requires a lot more trigonometry than in one dimension. It is the extraction of scalar relationships that requires some effort. The best way to learn is to see some examples.

### Newton's Second Law in Two Dimensions: on an Incline

Here is another example of Newton's Second Law. I show an approach to dealing with incline problems.

### Newton's Second Law in Two Dimensions: Hanging Mass

For this example, we look at a mass being hung by a series of ropes. I show another trick in dealing with Newton's Second Law problems, where the object is a vertex of ropes.

### Solution

1. A 750 kg mass is hanging from two cables, one is set 75 degrees from the horizontal and the other is set 60 degrees. What are the tensions in the cables?

### Solution

2. Using Newton's second law, derive the acceleration of an object sliding down a frictionless surface set an angle q above the horizontal.

### Solution

3. You think a pulley gives you leverage? Your car is stuck and there is a tree 100 m behind you. Instead of using a pulley, you attach a cable between the car and the tree and then pull on the cable in the middle the perpendicular direction with a 10,000 N tension force, T. If the cable is taught at a 2 degree angle (see figure), what is the force pulling on the car? 