Classical Mechanics
# Circular Motion

Alice and Bob are having fun throwing a ball to each other on a merry-go-round. Charlie looks at the game from outside of the merry-go-round. From his perspective, the ball thrown by Alice flies straight along the \(y\)-axis at a constant velocity of \(\vec v = v_0 \vec e_y\) to Bob. Bob can catch this ball after the flight time \(t_0 = T/4,\) because the merry-go-round has completed a quarter turn in the meantime.

But how does Alice observe (rotating reference system)? What average speed (average of absolute value of velocity vector) \[\overline{v'} = \frac{1}{t_0} \int_0^{t_0} |\vec v\,'| dt \] does the ball have from Alice's perspective?

Give the answer in units of \(v_0\) and with an accuracy of 3 decimal places.

\(\)

**Hints:**

- The merry-go-round rotates with constant frequency \(\omega = 2 \pi/T\), so that Alice \((\)point \(r_A)\) and Bob \((\)point \(r_B)\) move on circular paths in the stationary reference system. In the rotating reference system, both points \(r_A^\prime\) and \(r_B^\prime\) are stationary.
- Search for a \(2\times 2\) matrix \(\mathbf{D}\) such that the transformation reads \(\vec{r}^\prime = \mathbf{D} \cdot \vec{r}\). For the calculation of the average velocity, you may use the integral: \[ \int \sqrt{1 + x^2} d x = \frac{1}{2} \left( x \sqrt{1 + x^2} + \text{arcsinh}(x) \right).\]

**Bonus question:** Which (fictitious) forces act on the ball in the rotating reference system? How can we explain the path of the ball?

A rope loop of radius \(r = \SI[per-mode=symbol]{0.1}{\meter}\) and mass \(m = 50\pi\,\text{g}\) rotates on a frictionless table such that the tangential velocity of any point on the loop is \(v_\textrm{tangential}=\SI[per-mode=symbol]{6}{\meter\per\second}.\)

Find the tension in the loop (in Newtons).

A boy runs on a circular path of radius \( R=28\text{ meters}\) with constant speed \(u=4 \text{ m/s}\).

Another boy starts from the center of the circle to catch the first boy. The second boy always moves towards the first one. and maintains a speed of \(v = 4\text{ m/s}\). How long will the chase last in seconds?

**Details and Assumptions**:

1) Ignore air friction.

2) Take \(\pi = \dfrac{22}7\).

Consider a body, attached to a string of length \(r\), to be initially at rest in the vertical plane. It can be proved that if we give the body an initial velocity of \(v>\sqrt{5gr}\), then, the body can make complete revolutions in the vertical plane about the point of suspension.

Let us consider that the body, initially at rest, is given a velocity \(v<\sqrt{5gr}\). If at this value of \(v\) the body eventually hits the point of suspension then we can write:

\[ v^2 = (a+\sqrt{b})gr \; . \]

Find \(a\times b\).

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