INDIAN NATIONAL PHYSICS OLYMPIAD (INPhO) STAGE-II PREPARATORY TEST-1 Total Marks : 60
Total Time : 3 Hr.
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Question :
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Total
Marks :
5
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8
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13
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60
Score :
INPhO-1 - 1
INSTRUCTIONS: 1.
In this booklet consists of 7 questions.
2.
Questions consist of sub-questions. Write your detailed answer in seperate Answer booklet.
Computational tools such as calculators, mobiles, pagers, smart watches, slide rules, log tables etc. are not allowed. _____________________________________________________________________________________ 3.
1.
The space between a pair of coaxial cylindrical conductors is evacuated. The radius of the inner cylinder is a, and the inner radius of the outer cylinder is b, as shown in the figure below. The outer cylinder is called the anode, which may be give a positive potential V relative to the inner cylinder. A static homogenous
magnetic field B parallel to the cylinder axis, directed in to the plane of t he figure, is also present. Induced charges in the conductors are to be neglected. We study the dynamics of electrons with rest mass m and charge –e. The electrons are released at the surface of the inner cylinder.
[5 Marks]
(a)
First the potential V is turned on, but B = 0. An electron is set free with negligible velocity at the surface of the inner cylinder. Determine its speed v when it hits the anode. Give the answer both when a non-relativistic treatment is sufficient, and when it is not.
mc ev mc 2
Use this formula for relativistic t reatment c 1
2
Note : For the remaining parts of this porblem a non-relativistic treatment suffices. (b)
Now V = 0, but the homogeneous magnetic field B is present. An electron starts out with an initial
velocity v0 in the radial direction. For magnetic fields larger than critical v alue BC the electron will not reach the anode. Make a sketch of the trajectory of the electron when B is slightly more than BC . Determine BC .
Note : From now on both the potential V and the homogeneous magnetic field B are present. (c)
The magnetic field will give the electron a non - zero angular momentum L with respect to the cylinder axis. Write down an equation for the rate of change (dL/dt) of the angular momentum. Show that this equation implies that L – keBr 2 is constant during the motion, where k is a defined pure number. Here r is the distance of electron from t he cylinder axis. Determine the v alue of k.
(d)
Consider an electron, released from the inner cylinder with negligible velocity, that does not reach the anode, but has a maximum distance from the cylinder axis equal to rm . Determine the speed v of electron at the point where the radial distance is maximum, in terms of rm .
(e)
We are interested in using the magnetic field to regulate the electron current to the anode. For B larger than a critical magnetic field BC an electron, released with negligible velocity, will not reach the anode. Determine BC . INPhO-1 - 2
(f)
If the electrons are set free by heating the inner cylinder. Considering an electron will in general have an initial nonzero velocit y at the surface of the inner cylinder.
The component of the initial velocity parallel to B is vB , the components orthogonal to B are vr (in the radical direction) and v (in the azimuthal direction. i.e. orthogonal to the radial direction) Determine for this situation the critical magnetic fi eld BC for reaching the anode. 2.
Consider a long, solid, rigid, regular hexagonal prism like a common type of pencil as shown in f igure. The mass of the prism is M and it is uniformly distributed. The length of each side of the cross-sectional hexagonal is a. The moment of inertia I of the hexagonal prism about its central axis is I moment of inertia I ' about an edge of the prism is I '
17 12
5 12
Ma2 . The
2
Ma .
[5 Marks]
(a)
The prism is initially at rest with its axis horizontal on an inclined plane which makes a small angle with the horizontal figure. Assume that the surfaces of the prism are slightly concav e so that the prism only touches the plate at its edges. The effect of this concavity on the moment of inertia can be ignored. The prism is now displaced from rest and starts an uneven rolling down the plane. Assume that friction prevents any sliding and that the prism does not lose contact with the plane. The angular velocity just before a given edge hits the plane is i while f is the angular velocity immediately after the impact. Show that we may write f
(b)
The kinetic energy of the prism just before and after impact is similarly K i and K f . Show that we may write K f
( c)
s i and write the value of the coeffi cient s.
rK i and write the value of the coefficient r..
For the next impact to K i occur must exceed a minimum value K i ,min which may be written in the form
K i ,min
Mga
where g =9.81m/s2 is the acceleration of gravity. Find the coefficient in terms of the slope angle and the coefficient r..
INPhO-1 - 3
3.
A rigid cylindrical rod of radius R is held horizontal above the ground. With a string of negligible mass and length L L 2 R , a pendulum bob of mass m is suspended from point A at the top of the rod as shown in figure. The bob is raised until it is level with A and then released from rest when the string is taut. Neglect any stretching of the string. Assume the pendulum bob may be treated as a mass point and swings only in plane perpendicular to the axis of the rod. Accordingly, the pendulum bob is also referred to as the particle. The acceleration of gravity is
.
Let O be the origin of the coordinate system. W hen the particle is at point P, the string is tangential to the cylindrical surface at Q. The length of the l ine segment QP is called s. The unit tangent vector and the unit radial vector at Q are given by and counterclockwise from the vertical x-axis along OQ, is taken to be positive. When 0 , the length s is equal to L and the gravitational potential energy U of the particle is
zero. As the particle moves, the instantaneous time rate of change of and s are given by and s , respectively. Unless otherwise stated, all the speed and velocities are relative to the fixed point O.
[8 Marks]
The string is taut as the particle mov es. In terms of quantities introduced above ), find : (i.e., s, , R, L, g, t and r
4.
(a)
The relation between and s.
(b)
The velocity vQ of the moving point Q relative to O.
(c)
The particle’s velocity v relatives to O when it is at P..
(d) (e) (f)
The t - component of the particle’s acceleration relative to O when it is at P.. The particle’s gravitational potential energy U when it is at P. The speed v of the particle at the lowest point of its trajectory.
m
Watching a Rod in Motion
[8 Marks]
Physical situation : A pinhole camera, with the pinhole at x = 0 and at distance D from the x-axis, takes pictures of a rod by opening the pinhole for a very short time. There are equidistant marks along the x-axis by which the apparent length of the rod, as it is seen on the picture, can be determined form the pictures taken by the pinhole camera. On a pictures of the rod at rest, its length is L. However, the rod is not at rest, but is moving with constant velocity v along the x-axis.
INPhO-1 - 4
~
Basic relations : A picture taken by the pinhole camera shows a tiny segment of the rod at position x . (a)
What is actual position x of this segment at the time when the picture is taken? State your answer in ~
terms of x , D, L, v and the speed of light x = 3.00 × 10 8 ms –1. Employ the quantities
1 1 2
v and c
if any help to simplify your answer..
(b)
Find also the corresponding inverse relation, that is : express x in terms of x, D, L, v and c. Note : The actual position is the position in the frame in which the camera is at rest.
Apparent length of the rod : The pinhole camera takes a picture at t he instant when the actual position of the centre of the rod is at some point x 0 .
5.
(c)
In terms of the given variables, determine the apparent length of the rod on this picture.
(d)
Choose the correct one. The apparent length (i) increases first, reaches a maximum v alue, then decreases (ii) decreases first, reaches a minimum v alue, then increases (iii) decreases all the time (iv) increases all the time
A light elastic string on a smooth horizontal table has on of its ends fastened. The other end is attached to a particle of mass m. The string, of force constant k, is stretched to twice its natural length of l 0 and the particle is projected along the table at right angles to the string with a speed V 0 . [8 Marks] (a)
Show that in the subsequent motion, the string will attain its natural length again if its initial kinetic energy is less than a critical fraction of its initial total energy. Find this critical fraction.
(b)
If one-fifth of its initial energy is kinetic and if the string attains its natural length at some instant, describe the motion when the string becomes slack. Show that it wil l remain slack f or a duration Determine
6.
0 .
0 .
Kirchoff’s point rule states that the total current flowing into a junction of an electric circuit, consisting of resistors and batteries only, is equal to the total current flowing out of that junction. [13 Marks] (i) Give a brief physical explanation of Kirchoff’s point rule. A resistor circuit is constructed such that twelv e resistors are arranged to form a cube as shown in the figure.
Each resistor has a resistance of 2 . The eight points of the cube have been labelled A to H. A potential difference of 30 V is applied G in the diagram). (ii) Identify all groups of points that must be at the same voltage as each other. (iii) What is the magnitude of the current following from D to C ? We now replace two of the resistors with wires with zero resistance. In particular, we replace the resistor between A and B and the resistor between G and H. (iv) What is the potential difference between A and B ? (v) Identify all groups of points that must be at the same voltage as each other. (vi) What is the potential difference between the points C and G ? INPhO-1 - 5
7.
A metallic disc of radius R, thickness d and mass M is attached to a light, narrow conducting axle which passes through the disc’s centre. The disc is f ree to rotate and is totally immersed in a uniform constant magnetic field B which is perpendicular to the plane of the disc. [13 Marks] (a) Two electrical brushes are in contact with the axle and the rim of the disc. What potential difference must be applied across the terminals connected to the two brushes so that the rotational kinetic energy of the disc is T ? Assume that the disc is initially at rest. (b) Consider the case where the magnetic field B is confined to a small square area of size A 2 and average distance x(x < R) from the disc’s axis (see figure). If the disc’s electrical conductivity is , find an expression of the torque on the disc when it has a rotational kinetic energy of T.
INPhO-1 - 6
