A small body A starts sliding from the height h down an inclined groove passing into a half-circle of radius h 2 . Assuming the friction to be negligible, find the velocity of the body at the highest point of its trajectory (after breaking off the groove).

Here, in this problem, a body starts sliding, and it will leave the contact of the groove at some point. As in this problem, the body could not reach the uppermost point of the vertical track of radius r = h 2 , since the body does not gain a speed of 5 r g to complete the vertical circle. Thus, the particle A leaves the track at some point O with the speed v . We can get the velocity with the use of the principle of conservation of energy (Figure.). Now from the law of conservation of energy for the body A in the field of gravity, m g h - h 2 ( 1 + sin θ ) = 1 2 m v 2 or v 2 = g h ( 1 - sin θ ) . . . . . 1 From Newton's second law for the particle at the point O , F n = m w n N + m g sin θ = m v 2 h 2 . But, at the point O , the normal reaction N = 0 . So, v 2 = g h 2 sin θ . . . . . . 2 From the equation 1 , we will put the value of the velocity in the equation 2 , after solving the equation v = g h 3 , sin θ = 2 3 . After leaving the track at O , the particle A comes in the air and further goes up, and at a maximum height of its trajectory in the air, its the velocity (say v ' ) becomes horizontal (Figure.). Hence, the sought velocity of A at this point is, v ' = v cos ( 90 - θ ) = v sin θ = 2 3 g h 3 .

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A small disc $A$ slides down with initial velocity equal to zero from the top of a smooth hill of height $H$ having a horizontal portion. What must be the height of the horizontal portion $h$ to ensure the maximum distance $s$ covered by the disc? What is it equal to?

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.136.

A small body $A$ starts sliding from the height $h$ down an inclined groove passing into a half-circle of radius $\frac{h}{2}$ .

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Assuming the friction to be negligible, find the velocity of the body at the highest point of its trajectory (after breaking off the groove).

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.137.

A ball of mass

m

is suspended by a thread of length

l

. With what minimum velocity has the point of suspension to be shifted in the horizontal direction for the ball to move along the circle about that point? What will be the tension of the thread at the moment it will be passing the horizontal position?

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.138.

A horizontal plane supports a stationary vertical cylinder of radius $R$ and a disc $A$ attached to the cylinder by a horizontal thread $A B$ of length $l_{0}$ ( top view). An initial velocity $v_{0}$ , is imparted to the disc as shown in the figure. How long will it move along the plane until it strikes against the cylinder? The friction is assumed to be absent.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.139.

A smooth rubber cord of length $l$ , whose coefficient of elasticity is $k$ , is suspended by one end from the point $O$ . The other end is fitted with a catch $B$ . A small sleeve $A$ of mass $m$ starts falling from the point $O$ . Neglecting the masses of the thread and the catch, find the maximum elongation of the cord.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.140.

A small bar $A$ resting on a smooth horizontal plane is attached by threads to a point $P$ , and by means of a weightless pulley, to a weight $B$ possessing the same mass as the bar itself.

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Besides, the bar is also attached to a point $O$ by means of a light non-deformed spring of length $l_{0} = 50 cm$ , and stiffness $x = \frac{5 m g}{l_{0}}$ , where $m$ is the mass of the bar. The thread $P A$ having been burnt, the bar starts moving. Find its velocity at the moment when it is breaking off the plane.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.141.

A horizontal plane supports a plank with a bar of mass $m = 1.0 kg$ placed on it and attached by a light elastic non-deformed cord of length $l_{0} = 40 cm$ to a point $O$ . The coefficient of friction between the bar and the plank equals $k = 0.20$ . The plank is slowly shifted to the right until the bar starts sliding over it. It occurs at the moment when the cord deviates from the vertical by an angle $θ = 30 °$ . Find the work that has been performed by that moment by the friction force acting on the bar in the reference frame fixed to the plane.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.142.

A smooth light horizontal rod

A B

can rotate about a vertical axis passing through its end

A

. The rod is fitted with a small sleeve of mass

m

attached to the end

A

by a weightless spring of length

l_{0}

and stiffness

χ

. What work must be performed to slowly get this system going and reaching the angular velocity

ω

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.143.

A pulley fixed to the ceiling carries a thread with bodies of masses

m_{1}

and

m_{2}

attached to its ends. The masses of the pulley and the thread are negligible, friction is absent. Find the acceleration

\vec{w_{C}}

of the centre of inertia of this system.

A small disc A slides down with initial velocity equal to zero from the top of a smooth hill of height H having a horizontal portion. What must be the height of the horizontal portion h to ensure the maximum distance s covered by the disc? What is it equal to?

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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A small disc $A$ slides down with initial velocity equal to zero from the top of a smooth hill of height $H$ having a horizontal portion. What must be the height of the horizontal portion $h$ to ensure the maximum distance $s$ covered by the disc? What is it equal to?