Classical Mechanics

Circular Motion

Circular Motion - Level 4 Challenges

         

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 yy-axis at a constant velocity of v=v0ey\vec v = v_0 \vec e_y to Bob. Bob can catch this ball after the flight time t0=T/4,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) v=1t00t0vdt\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 v0v_0 and with an accuracy of 3 decimal places.


Hints:

  • The merry-go-round rotates with constant frequency ω=2π/T\omega = 2 \pi/T, so that Alice ((point rA)r_A) and Bob ((point rB)r_B) move on circular paths in the stationary reference system. In the rotating reference system, both points rAr_A^\prime and rBr_B^\prime are stationary.
  • Search for a 2×22\times 2 matrix D\mathbf{D} such that the transformation reads r=Dr\vec{r}^\prime = \mathbf{D} \cdot \vec{r}. For the calculation of the average velocity, you may use the integral: 1+x2dx=12(x1+x2+arcsinh(x)). \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 spherical ball of mass mm is kept at the highest point in the space between two fixed concentric spheres AA and BB . The smaller sphere AA has a radius RR and the space between the two spheres has a width dd. The ball has a diameter very slightly less than dd . All surfaces are frictionless. The ball is given a gentle push towards the right . The angle made by the radius vector of the ball with the upward vertical is θ\theta. What is the total normal reaction force exerted by the spheres on the ball in terms of angle θ\theta?

A rope loop of radius r=0.1 mr = \SI[per-mode=symbol]{0.1}{\meter} and mass m=50πgm = 50\pi\,\text{g} rotates on a frictionless table such that the tangential velocity of any point on the loop is vtangential=6 m/s.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 meters R=28\text{ meters} with constant speed u=4 m/su=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 m/sv = 4\text{ m/s}. How long will the chase last in seconds?

Details and Assumptions:

1) Ignore air friction.

2) Take π=227\pi = \dfrac{22}7.

Consider a body, attached to a string of length rr, to be initially at rest in the vertical plane. It can be proved that if we give the body an initial velocity of v>5grv>\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<5grv<\sqrt{5gr}. If at this value of vv the body eventually hits the point of suspension then we can write:

v2=(a+b)gr  . v^2 = (a+\sqrt{b})gr \; .

Find a×ba\times b.

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