Q # 1. Explain the difference between tangential velocity and the angular velocity. If one of these is given for a wheel of known radius, how will you find the other?
Ans.
Tangential velocity is the linear velocity of a particle moving along a curve
or circle. As the direction of the linear velocity is always along the tangent
to the circle, that is why it is called tangential velocity.
The
rate of change of angular displacement is called angular velocity. The
direction of angular velocity is along the axis of rotation of the body.
If
one of these two quantities are given for a wheel of known radius r, then we can find the
other by using the relation:
$\mathrm{v}=\mathrm{r} \omega$
Where
$v$ and $\omega$ are the tangential and angular velocity,
respectively.
Q # 2. Explain what is meant by
centripetal force and why it must be furnished to an object if the object is to
follow a circular path?
Ans.
The centripetal force is a force that makes a body follow a circular path. The centripetal force is always directed
towards the center of the circle.
The
direction of a body moving in a circular path is always changing. To bend the
normally straight path into circular path, a perpendicular force is needed,
called centripetal force.
Q # 3. What is meant by moment of
inertia? Explain its significance.
Ans.The product of mass of the particle and square of its perpendicular distance from the axis of rotation is called the moment of inertia. Mathematically, the moment of inertia $I$ is described as:
$$\mathrm{I}=\mathrm{m} r^{2}$$
Where
m is the mass of the particle and r is the perpendicular distance from the axis
of rotation.
The
moment of inertia plays the same role in angular motion as the mass in linear
motion.
Q # 4. What is meant by angular
momentum? Explain the law of conservation of angular momentum.
Ans.The
cross product of position vector and linear momentum of an object is known as
angular momentum.
The angular momentum L of a particle of mass m moving with velocity v and momentum p relative to the origin O is defined as:
$ \mathbf{L}=\mathbf{r} \times \mathbf{p} $
By solving the above mentioned expression, the magnitude of angular momentum can be determined equal to product of moment of inertia and angular velocity:
$\mathrm{L}=\mathrm{I} \omega$
Law of Conservation of Angular Momentum
Statement: If no external torque acts on a system, the total angular momentum of the system remains constant.
$I_{1} \omega_{1}=I_{2} \omega_{2}=\text { constant }$
$Q$ \# 5.Show that orbital angular momentum $L_{0}=$ mvr.
Ans. The angular momentum \(L_{o}\) of a particle of mass \(m\) moving with velocity \(v\) and momentum \(p\) relative to the origin \(O\) is defined as:
\[L_{o} = r \times p\]
\[L_{o} = rp\ \sin\ \theta\ \widehat{n}\]
The magnitude of angular momentum will be:
\[L_{o} = rp\ sin\ \theta\]
Because \(\ p = mv\)
\[L_{o} = mvr\ sin\ \theta\]
Since the angle between radius \(r\) and tangential velocity \(v\) is \(90^{o}\), so
\[L_{o} = mvr\ \sin\ 90^{o} = mvr\]
Hence proved.
Q # 6. Describe what should be the minimum velocity, for a satellite, to orbit close to earth around it.
Ans. Consider a satellite of mass \(m\) is moving in a circle of radius \(R\) around the earth. In circular orbit for a low flying satellite, the centripetal acceleration is provided by the gravity.
\[g = \frac{v^{2}}{R}\]
where\(v\) it the tangential velocity of the satellite. Solving equation (1), we have:
\[v = \sqrt{gR}\]
Near the surface of the earth, the gravitational acceleration \(g = 9.8\ ms^{- 1}\)and \(R = 6.4 \times 10^{6}m\).
\[\Longrightarrow v = \sqrt{9.8 \times 6.4 \times 10^{6}} = 7.9\ kms^{- 1}\]
This is the minimum velocity necessary to put a satellite into the orbit and is called critical velocity.
Q # 7. State the direction of following vectors in simple situations; angular momentum and angular velocity.
Ans. The directions of angular momentum and angular velocity are used to described by right hand rule:
Grasp the axis of rotation in right hand with the figures curling in the direction of rotation, then the erected thumb will give the direction of angular velocity and angular momentum.
Q # 8. Explain why an object, orbiting the earth, is said to be freely falling. Use your explanation to point out why objects appear weightless under certain circumstances.
Ans.An object is put into an orbit around the earth will move in a curved path under the action gravity. The curvature of the path is such that it matches the curvature of the earth, and object does not touch the earth surface. As the object continues to fall around the Earth, so it is said to be freely falling object.
When a body is falling freely, it moves with an acceleration g. And the bodies falling with acceleration g appear weightless.
Q # 9. When mud flies off the tyre of a moving bicycle, in what direction does it fly? Explain.
Ans.When the mud flies off the tyre of a moving bicycle, it always flies along the tangent to the tyre. This is due to the reason that the linear velocity is always tangent to the circle, and the mud will fly in the direction of linear velocity.
Q # 10. A disc and a hoop start moving down from the top of an inclined plane at the same time. Which one will have greater speed on reaching the bottom?
Ans. The formulae for the velocity of the disc and the hoop are given by:
\(v_{disc} = \sqrt{\frac{4}{3}gh}\) and \(v_{hoop} = \sqrt{gh}\)
So it is clear from the above relations that the disc will be moving with greater speed on reaching the bottom.
Q # 11. Why a diver does changes its body position before and after diving in the pool?
\textbf{Ans.}When the diver jumps from the diving board, his legs and arm are fully extended. The diver has large moment of inertia \(I_{1}\)but the angular velocity \(\omega_{1}\) is small. When the diver curls his body, the moment of inertia reduces to \(I_{2}\). In order to conserve the angular momentum, the value to angular velocity increases to \(\omega_{2}\) such that
\[I_{1}\omega_{1} = I_{2}\omega_{2} = constant\]
In this way, the diver can make more somersaults before entering the water.
Q # 12. A student holds two dumb-bells with stretched arms while sitting on a turn table. He is given a push until he is rotating at certain angular velocity. The student then pulll the dumb bells towards his chest. What will be the effect on rate of rotation?
\textbf{Ans.}Initially, the arms of the students are fully extended, so he has large moment of inertia \(I_{1}\) but angular velocity \(\omega_{1}\) is small. When the student curls his body, the moment of inertia reduces to \(I_{2}\). In order to conserve the angular momentum, the value of angular velocity increases to \(\omega_{2}\) such that
\[I_{1}\omega_{1} = I_{2}\omega_{2} = constant\]
Thus the rate of rotation will increase.
Q # 13. Explain how many minimum number of geo-stationary satellites are required for global coverage of TV transmission.
Ans. The total longitude of earth is \(360{^\circ}\) and one geostationary satellite covers of longitude \(120{^\circ}\). So the whole earth can be covered by three correctly positioned geostationary satellites.
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