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# Circular Motion

The wheels on the bus go round and round, but can you name all the forces in a rotating reference frame? Learn to derive these and more through sheer force of reason in 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 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.

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 $$a\times b$$.

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 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 '?

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