Two concave mirrors of equal radii of curvature $$R$$ are fixed on a stand facing opposite directions. The whole system has a mass $$m$$ and is kept on a frictionless horizontal table. Two blocks $$A$$ and $$B$$, each of mass $$m$$, are placed on the two sides of the stand. At $$t = 0$$, the separation between $$A$$ and the mirrors is $$2 R$$ and also the separation between $$B$$ and the mirrors is $$2 R$$. The block $$B$$ moves towards the mirror at a speed $$v$$. All collisions which take place are elastic. Taking the original position of the mirrors stand system to be $$x = 0$$ and X-axis along $$AB$$, find the position of the images of $$A$$ and $$B$$ at $$t$$- a) $$\dfrac{R}{v}$$ b) $$\dfrac{3R}{v}$$ c) $$\dfrac{5R}{v}$$
a) In time $$t=R/V$$ the mass $$B$$ must have moved $$(v\times R/v)=R$$ closer to the mirror stand
So, For the block $$B$$:
$$u=-R,f=-R/2$$
$$\therefore \dfrac {1 }{ v } =\dfrac { 1 }{ f } -\dfrac { 1 }{ u } =-\dfrac {2 }{ R } +\dfrac { 1 }{ R } =-\dfrac { 1}{ R} $$
$$\Rightarrow v=-R$$ at the same place.
For the block $$A$$ : $$u=-2R,f=-*R/2$$
$$\therefore \dfrac {1 }{ v } =\dfrac { 1 }{ f } -\dfrac { 1 }{ u } =-\dfrac {2 }{ R } +\dfrac { 1 }{ 2R } =-\dfrac { -3}{ 2R} $$
$$\Rightarrow v=\dfrac { -2R }{ 3 }$$ image of $$A$$ at $$\dfrac { 2r }{ 3 } $$from PQ in the x-direction.
So, with respect to the given coordination system,
$$\therefore$$ Position of $$A$$ and $$B$$ are $$\dfrac {-2R }{ 3} $$, $$R$$ respectively from origin
b) When $$t=3R/v$$, the block $$B$$ after colliding with mirror stand must have come to rest (elastic collision) and the mirror have travelled a distance $$R$$ towards left form its initial position.
So, at this point of time.
For block $$A$$:
$$u=-R,f=-R/2$$
Using lens formula, $$v=-R$$ (from the mirror)
So, position $$x_A=-2R$$ (from origin of coordinate system)
For block $$B$$:
Image is at the same place as it is $$R$$ distance from mirror. Hence, position of image is $$'0'$$.
Distance from $$PQ$$ (coordination system)
$$\therefore$$ positions of images of $$A$$ and $$B$$ are $$=-2R,0$$ from origin.
c) Similarly, it can be proved that at time $$t=-5R/v$$
the position of the blocks will bw $$-3R$$ and $$-4R/3$$ respectively.
An object $$2.4\ m$$ in front of a lens forms a sharp image on a film $$12\ cm$$ behind the lens. A glass plate $$1\ cm$$ thick of refractive index $$1.50$$ is interposed between lens and film with its plane faces parallel to film. At what distance (from lens) should object be shifted to be in sharp focus on film?
An object is placed at a distance of $$2f $$ from a concave mirror. Light reflected from the mirror falls on a plane mirror. The distance of the plane mirror from the concave mirror equals $$f$$. The distance of the final image from the concave mirror (due to reflection at both concave and plane mirror) is :
A small object is placed $$50\ cm$$ to the left of a thin convex lens of focal length $$30\ cm$$ . A convex spherical mirror of radius of curvature $$100\ cm$$ is placed to the right of the lens at a distance of $$50\ cm$$. The mirror is titled such that the axis of the mirror is at a angle $$\displaystyle \theta ={ 30 }^{ \circ }$$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates $$(in\ cm)$$ of the point $$(x, y)$$ at which the image is formed are:
A biconvex lens of focal length 15 cm is in front of a plane mirror. The distance between the lens and the mirror is 10 cm. A small object is kept at a distance of 30 cm from the lens. The final image is :
A diverting lens of focal length $$20 cm$$ and converging mirror of focal length $$10 cm$$ are placed coaxially at a separation of $$5 cm$$. Where should an object be placed so that a real image is found at the object itself?
shows a transparent hemisphere of radius $$3.0cm$$ made of a material of refractive index $$2.O$$. (a) A narrow beam of parallel rays is incident on the hemisphere as shown in the figure. Are the rays totally reflected at the plane surface? (b) Find the image formed by the refraction at the first surface. (c) Find the image formed by the reflection or by the refraction at the plane surface. (d) Trace qualitatively the final rays as they come out of the hemisphere.
A diverging lens of focal length $$20 cm$$ and a converging lens of focal length $$30 cm$$ are placed $$15 cm$$ apart with their principal axes coinciding. Where should an object be placed on the principal axis so that its image is formed at infinity ?
Two convex lenses, each of focal length $$10 cm$$, are placed at a separation of $$15 cm$$ with their principal axes coinciding, (a) Show that a light beam coming parallel to the principal axis diverges as it comes out of the lens system. (b) Find the location of the virtual image formed by the lens system of an object placed far away. (c) Find the focal length of the equivalent lens. (Note that the sign of the focal length is positive although the lens system actually diverges a parallel beam incident on it).
The combination of convex lens and concave lens each of focal length $$10\ cm$$ when combines, behave as :
Three-lens systems. In Figure, stick figure O (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to O, which is at object distance $$p_1$$. Lens 2 is mounted within the middle boxed region, at distance $$d_{12}$$ from lens 1. Lens 3 is mounted in the farthest boxed region, at distance $$d_{23}$$ from lens 2. Each problem in Table refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by C for converging and D for diverging; the number after C or D is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance $$i_3$$ for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification M for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object O or noninverted (NI), and (e) on the same side of lens 3 as object O or on the opposite side.