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Classical Mechanics

Circular Motion

Circular Motion - Level 4 Challenges

         

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

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 frictionless track consists of a horizontal part of unknown length, which connects to a vertical semicircle of radius r . An object, which is given an initial velocity v , is to move along the track in such a way that after leaving the semicircle at the top it is to fall back to its initial position. What should the minimum length of the horizontal part be '?

A spherical ball of mass \(m\) is kept at the highest point in the space between two fixed concentric spheres \(A\) and \(B\) . The smaller sphere \(A\) has a radius \(R\) and the space between the two spheres has a width \(d\). The ball has a diameter very slightly less than \(d\) . 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 \(\theta\). What is the total normal reaction force exerted by the spheres on the ball in terms of angle \(\theta\)?

A small block can move in a straight horizontal line \(AB\).Flash lights from one side projects its shadow on a curved vertical wall which has a horizontal cross section as a circle. If tangential and normal acceleration of shadow of the block on the wall as a function of time can be represented as:

\[a_N=\frac{cvR}{(2Rt-vt^2)} \quad , \quad a_T=\frac{R(vt-R)v^{\frac{d}{e}}}{(2Rt-vt^2)^{\frac{f}{g}}}\].

Find \(|c|+|d|+|e|+|f|+|g|\)


Details and Assumptions

  • As shown in figure \(v\) is the constant along \(AB\).

  • Given Figure is top view of the setup.

  • \(c,d,e,f,g\) are natural numbers and \(\frac{d}{e} ,\frac{f}{g}\) are in simplest forms.

  • \(a_N,a_T \) represent normal and tangential accelerations respectively.

  • you can neglect minus sign in accelerations due to directions.

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