Newton's laws - Pendulum in an accelerating car

Problem Statement:

A teddy bear is suspended from the ceiling of a car by a rope (ignore the mass of the rope). Determine the acceleration of the car if the rope makes an angle α with the vertical.

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Solution:

When a car moves with an acceleration a (which we will assume constant), the rope that supports the teddy bear will tilt in the opposite direction to the motion to form an angle α with the vertical.

The following figure shows the car seen in a reference frame at rest on the ground. The diagram shows the forces acting on the teddy bear as well as the positive direction of the Cartesian axes. As the teddy bear is suspended from the ceiling by a rope, it is subject to a tension force from it. The weight also acts on the doll because it is close to the Earth and therefore experiences its gravitational force.

The teddy bear has a uniformly accelerated motion with acceleration a with respect to the reference frame at rest on the ground (the same as that of the car). We have represented the projections of the forces on the axes in the following figure.

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To determine the acceleration of the car we must apply Newton’s second law to the motion of the teddy bear:

The projection on the Cartesian axes is:

And using the properties of complementary angles we get:

Or:

And finally by dividing the two equations we get:

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