Example in the book: A simple accelerometer

Question

A simple accelerometer

You tape one end of a piece of string to the ceiling light of your car and hang a key with mass m to the other end (Figure 5.7). A protractor taped to the light allows you to measure the angle the string makes with the vertical. Your friend drives the car while you make measurements. When the car has a constant acceleration with magnitude a toward the right, the string hangs at rest (relative to the car), making an angle $B$ with the vertical.

• (a) Derive an expression for the acceleration $a$ in terms of the mass m and the measured angle $B$.

• (b) In particular, what is a when $B$ = 45? When $B$ = 0?

I don't care about the answers, the important thing is the following:-

The book says The string and the key are at rest with respect to the car, but car, string, and key are all accelerating in the +x direction. Thus, there must be a horizontal component of force acting on the key.

That's the reason the book decided to consider a force in the $+x$ direction, but I'm looking for a better explanation: how would I find detect the force in the $+x$ direction in another way? To me, when I draw the free body diagram of the string, there looks to be no force acting on the $+x$ direction! I understand it starts with noticing that the string is attached to the ceiling of the car, and that the car has force causing acceleration in one direction, but I don't know how to go further than that.

Answer 1

Acceleration and force are not the same thing.

So first simplify the problem by just talking about acceleration.

For example, suppose the car is accelerating forward at a rate of 1.0 meter per second per second. It also feels an "acceleration" due to gravity, which is 9.8 m/(s^2). If you want to know the angle at which the key hangs, it is the angle whose tangent is 1.0/9.8. Hopefully that is enough for you to run with.

Then if you want to know force, make use of f=ma. i.e. multiply the acceleration by the vehicle's mass.

Answer 2

Since the net acceleration is in the +x direction, the x component of the tension in the string must be providing the net force, since the only other force that applies, gravity, has no x component. Meanwhile, the y component of the string tension cancels the y acceleration from gravity. There is no other force acting on the key.

Finally, I believe there is a typo in the statement of the question. Your expression for the acceleration will include the gravitational acceleration g but not the mass of the key.

Answer 3

Did you forget to include the D'Alembert forces in the Free Body Diagram?

http://en.wikipedia.org/wiki/D'Alembert's_principle