B M Sharma Solutions for Chapter: Circular Motion, Exercise 55: CONCEPT APPLICATION EXERCISE 9.4

A simple pendulum of length $l$ and mass $m$ free to oscillate in vertical plane. A nail is located at a distance $d=l-a$ vertically below the point of suspension of a simple pendulum. The pendulum bob is released from the position where the string makes an angle of $90°$ from vertical. Discuss the motion of the bob if

(a) $l=2a$

(b) $l=2.5a$.

A ball is attached to a horizontal cord of length $L$ whose other end is fixed,

(a) If the ball is released, what will be its speed at the lowest point of its path?

(b) A peg is located a distance $h$ directly below the point of attachment of the cord. If $h=0.75L$, what will be the speed of the ball when it reaches the top of its circular path about the peg?

A particle is suspended from a fixed point by a string of length $5 \mathrm{m}$. It is projected from the equilibrium position with such a velocity that the string slackens after the particle has reached a height $8 \mathrm{m}$ above the lowest point. Find the velocity of the particle just before the string slackens. Find also to what height the particle can rise further.

A small box of mass $m$ is kept on a fixed, smooth sphere of radius $R$ at a position where the radius through the box makes an angle of $30°$ with the vertical. The box is released from the position.

(a) What is the force exerted by the sphere on the box just after the release?

(b) Find the distance travelled by the box before it leaves contact with the sphere.

Figure shows a smooth track, a part of which is a circle of radius $r$. A block of mass $m$ is pushed against a spring of spring constant $k$ fixed at the left end and is then released. Find the initial compression of the spring so that the block presses the track with a force mg when it reaches the point $P$, where the radius of the track is horizontal.

A bob of mass $m$ is projected with a horizontal velocity $v=\sqrt{gl/2}$ as shown in the figure. In consequence, it moves in a circular path in a vertical plane by the inextensible string which passes over the smooth fixed peg. Find the maximum angle that the bob swings in the left hand side.

A block is projected with a speed $v_0$ such that it strikes the point of projection after describing the path as shown by the dotted line. If friction exists for the path of length $d$ and the vertical circular path is smooth, assuming $\mu =$ coefficient of friction

(a) Find $v_0$.

(b) What is the minimum value of $v_1$.

A small ball is suspended from point $O$ by a thread of length $l$. A nail is driven into the wall at a distance of $1/2$ below $O$, at $A$. The ball is drawn aside so that the thread takes up a horizontal position at the level of point $O$ and then released.

(a) At what angle from the vertical of the ball’s trajectory, will the tension in the thread disappear?

(b) What will be the highest point from the lowermost point of circular track, to which it will rise?