  A simple accelerometer is constructed inside a car by suspending an object of mass m from a string of length L that is tied to the car's ceiling. As the car accelerates the string–object system makes a constant angle of θ with the vertical.(a) Assuming that the string mass is negligible compared with m, derive an expression for the car's acceleration in terms of θ and show that it is independent of the mass m and the length L. (Use the following as necessary: θ and g.)(b) Determine the acceleration of the car when θ = 22.6°.

Question

A simple accelerometer is constructed inside a car by suspending an object of mass m from a string of length L that is tied to the car's ceiling. As the car accelerates the string–object system makes a constant angle of θ with the vertical.
(a) Assuming that the string mass is negligible compared with m, derive an expression for the car's acceleration in terms of θ and show that it is independent of the mass m and the length L. (Use the following as necessary: θ and g.)
(b) Determine the acceleration of the car when θ = 22.6°.

Step 1

As per the question, an accelerometer is constructed by a string of length L and mass of the object is m which is connected with the string. String object system makes an angle θ with vertical.

a) String mass is negligible with respect to block, we have to derive expression’s for car.

b) We have to determine the acceleration of the car when string makes an angle θ=22.6 with vertical.

Step 2

When the car accelerates with certain acceleration then object which is hanged with the string will experiences a pseudo force due to which it will move opposite to the direction of the motion of the car.

So, string will incline at an angle θ with the vertical due to the acceleration of the car.

Step 3

Now, applying the force balance in the vertical and the horizontal direction.
Let the tension dev...

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