# 'Simple' Pendulum in Hanging in a Car Udergoing Circular Motion

1. Jun 16, 2010

### modulus

1. The problem statement, all variables and given/known data
A simple pendulum (massless string, neglect friction) is supended from the ceiling of a car taking a turn of radius 10m at a speed of 36km/h. Find the angle made by the string of the pendulum wih the vertical if this angle does not change during the turn. Take g=10ms-2

2. Relevant equations
The centripetal acc of the car (and the bob of the pendulum too, aince it moves along with the car)= v2/r

3. The attempt at a solution
My instinct told me that the pendulum's bob should move away from the centre of the car's circle of motion. But, the bob would also move in a circle along with car, which implies that there is a centripetal force on the bob. I think the centripetal acceleration will be provided by the tension in the string.

So, I equated the tension to the component of gravitation along the string:
T = mg secѳ

And I took the component of tension which pointed towards the center of the circle of motion of the car, and equated it to the centripetal acceleration on the bob:
T sinѳ = mv2/r

Combining the two, I got:
mg tanѳ = mv2/r
ѳ = 45®

My answer was right, but what I couldn’t understand was why the pendulum would move away from the center of the car’s circle of motion. What compells the string of the bob to do so. There wasn’t any force acting on the bob away from thecenter of the circle.

And, that question has been bugging me a lot. I think my method might also be wrong, because I can’t even justidy my opening statement to solve the problem.

2. Jun 16, 2010

### hikaru1221

(1): In the reference frame of the car, the bob experiences the centrifugal force (equal to centripetal force in value, opposite in direction). It is a fictitious force which arises from that the car is a non-inertial frame. Because of the presence of the centrifugal force, the bob moves away from its initial position until equilibrium is formed when the forces cancel out each other.

(2): In the reference frame of the ground, when the car starts to accelerate, the point where the string is held begins to move with different velocity from the bob. Because of that, the two ends of the string are no longer in the same vertical line, and thus, again, the bob moves away from its initial position.

So what makes the string to do so is that there is relative motion between the bob and the car:
_ In (1), the bob is not at rest in the frame of the car.
_ In (2), the relative motion of the ends implies the relative motion of the bob and the car, though they're not the same.

Last edited: Jun 16, 2010
3. Jun 16, 2010

### collinsmark

The answer lies in Newton's first and second laws. Consider part of the first law to start. An object in motion tends to stay in motion unless acted upon by an outside force. With that in mind, consider the following questions:

(I) Before the car starts turning, when car is traveling in a straight line (but still in motion), and the bob and the string and the car are in a state of equilibrium, with nothing accelerating, is the bob also in motion? (Hint: If you claim that the bob is not in motion, even though the car is, and since the bob and the car are not moving relative to each other, you'll have a lot of explaining to do.)

(II) According to Newton's second law, if a force acts upon the car, the car will accelerate. If the force is perpendicular to the car's velocity, the car will accelerate in a circle. If no forces act upon the bob at this time, the bob will continue to go in a straight line (according to Newton's first law), and the bob will fly right out the car's window (in a straight line) as the car turns. So if the bob is turning in a circle too along with the car, what force causes the bob to accelerate?

(III) We've established that the car is accelerating. If the top of the string is attached to the car, is the top of the string also accelerating?

(IV) If the top of the string is accelerating, what happens to the bottom of the string, and the bob?

4. Jun 16, 2010

### modulus

Gosh, thanks.

You guys really explained that well. Really well.

After the top of the string moves with the car, the tension on the bob changes direction, and the bob accelerates (moves in a circle) along with the car when equilibrium is established(Right?).

5. Jun 16, 2010

There ya go.