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Chapter 4: Problem 42

A conical pendulum consists of a simple pendulum moving in a horizontal circle as shown in the figure, \(C\) is the pivot. \(O\) the centre of the circle in which the pendulum bob moves and \(\omega\) the constant angular velocity of the bob. If \(\vec{L}\) is the angular momentum about point \(C\), then (1) \(\vec{L}\) is constant (2) only direction of \(\vec{L}\) is constant (3) only magnitude of \(\vec{L}\) is constant (4) none of the above

Short Answer

Expert verified
(1) \(\vec{L}\) is constant.

Step by step solution

01

Identify the properties of angular momentum

The angular momentum {L}9 of an object rotating about a point is given by {L} =  {r}  { imes}  {p}, where {r} is the position vector from the point to the object, and {p} = m ext{{}}{v}  is the linear momentum of the object, with {m} as the mass and {v} as the velocity of the object.
02

Express angular momentum in terms of angular velocity

For a conical pendulum moving in a horizontal circle with angular velocity {}, the angular momentum {L} = m ({r {v}) = m ({r}{2}{}), where {r} is the radius of the circle and {v} is the tangential velocity of the pendulum bob, equal to {r}{}.
03

Determine the direction of angular momentum

The direction of angular momentum is given by the right hand rule and is perpendicular to the plane of rotation. For a conical pendulum moving in a horizontal circle, {} points vertically upward if the circle rotates counter-clockwise.
04

Evaluate variations in angular momentum

For a conical pendulum, both the magnitude and the direction of the angular momentum {} are constant since the angular velocity {} is constant, ensuring consistency in rotational motion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conical Pendulum
A conical pendulum is a fascinating physical system where a weight, or bob, affixed to one end of a string, moves in a circular motion while the string makes an angle with the vertical. This configuration allows the pendulum to trace out a cone during its motion. The point where the string is fixed is referred to as the pivot, and in this system, the weight or bob traces a horizontal circle around this fixed point.
Understanding the conical pendulum requires recognizing that even though the bob moves in a two-dimensional circular path, the pendulum itself spans three dimensions because of its vertical displacement.
  • The tension in the string provides the necessary centripetal force to keep the bob moving in a circle.
  • Gravity acts perpendicularly to this centripetal force, acting downwards.
  • The combination of these forces allows for the stable circular motion of the pendulum bob.
Angular Velocity
Angular velocity is a measure of how quickly an object rotates or revolves around a particular axis. In the context of a conical pendulum, angular velocity denoted by \(\omega\), is a constant value that describes how fast the pendulum bob moves along its circular path.
It's important to understand that the angular velocity in a conical pendulum determines the speed of revolution without altering the radius of the circle; \(\omega\) remains unchanged as long as the forces acting on the bob remain constant.
Some key ideas about angular velocity include:
  • It is calculated as the angle turned, measured in radians, over time.
  • In a circular path, angular velocity provides insights into how quickly the angle relative to a start position changes.
  • A consistent angular velocity means the pendulum bob moves at a steady pace, ensuring a stable motion.
Rotational Motion
Rotational motion refers to the motion of an object around a circular path. In the case of a conical pendulum, the motion is unique due to its conical trace, where the bob maintains a horizontal circular motion as the pendulum rotates. This consistent circular path is guided by the interplay of gravitational force and tension within the string.
The rotational motion of a conical pendulum is completely characterized by the constancy of both angular velocity and the tension in the string. Without these two elements balancing out the gravitational pull, the pendulum would not retain its consistent circular path.
Several factors defining rotational motion are:
  • The centripetal force needed to keep the object moving in a circle is provided by the tension in the string.
  • Any change in the radius or tension can alter the rotational trajectory.
  • The torque experienced, due to the pivot, affects the angular momentum, making rotational motion a complex yet fascinating topic of study.

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Most popular questions from this chapter

A ring of mass \(3 \mathrm{~kg}\) is rolling without slipping with linear velocity \(1 \mathrm{~ms}^{-1}\) on a smooth horizontal surface. A rod of same mass is fitted along its one diameter. Find total kinetic energy of the system (in J).

A particle of mass ' \(m\) ' is rigidly attached at ' \(A\) ' to a ring of mass ' \(3 m\) ' and radius ' \(r\) '. The system is released from rest and rolls without sliding. The angular acceleration of ring just after release is (1) \(\frac{g}{4 r}\) (2) \(\frac{g}{6 r}\) (3) \(\frac{g}{8 r}\) (4) \(\frac{g}{2 r}\)

A horizontal disc rotates freely about a vertical axis through its centre. A ring, having the same mass and radius as the disc, is now gently placed on the disc. After some time, the two rotate with a common angular velocity, then (1) some friction exists between the disc and the ring (2) the angular momentum of the 'disc plus ring' is conserved (3) the final common angular velocity is \((2 / 3)\) rd of the initial angular velocity of the disc (4) (2/3)rd of the initial kinetic energy changes to heat

A metal plate which is square in shape of side length \(l=2 \mathrm{~m}\) has a groove made in the shape of two quarter circles joining at the centre of the plate. The plate is free to rotate about vertical axis passing through its centre. The moment of inertia of the plate about this axis is \(I_{0}=4 \mathrm{~kg}-\mathrm{m}^{2}\). A particle of mass \(m=1 \mathrm{~kg}\) enters the groove at end \(A\) travelling with a velocity of \(v_{A}=2 \mathrm{~m} / \mathrm{s}\) parallel to the side of the square plate. The particle move along the frictionless groove of the horizontal plate and comes out at the other end \(B\) with speed \(v\). Find the magnitude of \(v\) (in \(\mathrm{m} / \mathrm{s}\) ) assuming that width of the groove is negligible.

A slender rod of mass \(M\) and length \(L\) rests on a horizontal frictionless surface. The rod is pivoted about one of its ends. The impulse of the force exerted on the rod by the pivot when the rod is struck by a blow of impulse \(J\) perpendicular to the rod at other end is (l) \(J\) (2) \(\frac{J}{2}\) (3) \(\frac{J}{3}\) (4) information is insufficient
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