Chapter test-3-physics-unit and dimension

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Q 1. A B C D The velocity v (in cm/sec) of a particle is given 
        in terms of time t (in sec) by the equation,

        v = at + $\frac{b}{t+c}$The dimensions of a, 
        b and are:  (CPMT 1990)                                                                                                            

                      a                                        b                             c

               (a) L²                                    T                              LT²

                (b) LT²                                                             LT                          L

                (c) LT^-2                                 L                             T

                (d) L                                     LT                          T²

               

Q 2.  A B C D If V denotes the potential difference across the plates of a capacitor C, the dimensions of CV² are:

(a) not expressible in MLT                            
(b) MLT^-2            
                         

(c) M²LT^-1    
                                         

(d) ML²T^-2          

Q 3. A B C D If L denotes the inductance of an inductor through which a current I is flowing, the dimensions of LI² are:

(a) ML²T^-2           
                 
(b) not expressible in MLT

(c) MLT^-2     
                         
(d) M²L²T^-2 

Q 4. A B C D According to quantum mechanics light travels in the form of packets and energy associated with each packet is E = hv, where h is Planck's constant and v is the frequency. The dimensional formula for Planck's constant is:

(a) [M¹L²T^-1]                       
(b) [M²L^-1 T²]      
(c) [ML²T^-1]                         
(d) [ML^-2T¹]

Q 5. A B C D Using dimensional analysis which of the following is correct (m is relativistic mass, m₀ is rest mass, V is the velocity of particle and c is the velocity of light)?

                   (a) m = $\frac{{{m}_{0}}}{\sqrt{1-\frac{{{V}^{2}}}{{{c}^{2}}}}}$    

         (b) m = $\frac{{{m}_{0}}}{\sqrt{1-{{V}^{2}}}}$     

         (c) m = $\frac{{{m}_{0}}}{\sqrt{1-{{c}^{2}}{{V}^{2}}}}$           

         (d) m = $\frac{{{m}_{0}}}{\sqrt{1-{{c}^{2}}}}$

Q 6. A B C D A spherical body of mass m and radius r is allowed to fall in a medium of viscosity h. The time in which the velocity of the body increases from zero to 0.63 times, the terminal velocity (v) is called time  constant (t).  Dimensionally t can be represented by:

                   (a) $\frac{m{{r}^{2}}}{6\pi \eta }$            
                 
          (b) $\sqrt{\left( \frac{6\pi mr\eta }{{{g}^{2}}} \right)}$   

          (c) $\frac{m}{6\pi \eta rv}$               
                 
          (d) none of these

               

Q 7. A B C D The rotational kinetic energy is given as KE = $\frac{1}{2}I{{\omega }^{2}}$then dimensional formula for moment of inertia is:

(a) [ML²T^-2]        
(b) [ML^-1 T^-1]                       
(c) [ML²T⁰]                          
(d) [M^-1L²T^-1]      

Q 8. A B C D In the relation:  y = a sin (wt – Kx) the dimensional formula for K is:

(a) M⁰LT               
(b) M⁰L^-1T⁰                          
(c) M⁰LT^-1             
(d) M⁰L^-1 T^-1        

Q 9. A B C D In the relation:  y = a cos (wt + Kx) the dimensional formula for Kx is same as that of:

(a) a/w                  
(b) a/y                                  
(c) wt/a               
(d) ya/wt              

Q 10. A B C D A student writes four different expressions for the displacement ‘y’ in a periodic motion:

(1) y = a sin $\frac{2\pi t}{T}$      
(2) y = a sinVt                     
(3) y = $\frac{a}{T}$sin $\frac{t}{a}$       
(4) y = $\frac{a}{\sqrt{2}}\left[ \sin \frac{2\pi t}{T}+\cos \frac{2\pi t}{T} \right]$

                where 'a' is maximum displacement, V is the speed and T is the time period; then dimensionally:

(a) 1 and 2 are wrong     
(b) 2 and 3 are wrong     
(c) 3 and 4 are wrong      
(d) 4 and 1 are wrong 

               

Q 11. A B C D In the relation: $\frac{dy}{dt}$ = 2w sin (wt + f₀) the  dimensional formula for (wt + f₀) is :

                   (a) MLT                                
         (b) MLT⁰                                      
         (c) ML⁰T⁰                             
         (d) M⁰L⁰T⁰

               

Q 12. A B C D In the relation:  x = R cos (wt – Kx) the dimensional                       formula for wt of is same as that for:

(a) x/R                                  
(b) Kx/R                               
(c) Kw/xt             
(d) wR/K              

               

Q 13. A B C D Given that g is acceleration due to gravity and R is the radius of the earth. Then [g/R]^1/2 possesses the dimensions of:

(a) orbital speed               
(b) angular speed            
(c) escape velocity           
(d) time period      

Q 14. A B C D The frequency of vibration of a string is given by:v = $\frac{p}{2l}{{\left[ \frac{F}{m} \right]}^{1/2}}$ 
       

Here p is the number of segments in which the string is divided, F is the tension in the string and l is its length. The dimensional formula for m is:

(a) M⁰L⁰T⁰            
(b) ML^-1T⁰                            
(c) ML⁰T^-1             
(d) M⁰LT^-1

Q 15. A B C D Given that tan q = v²/rg gives the angle of banking of the cyclist going round the curve. Here v is the speed of cyclist, r is the radius of the curve and g is acceleration due to gravity. Which of the following statements about the relation is true?

(a) It is both dimensionally as well as numerically correct                              

(b) It is neither dimensionally correct nor numerically correct                   

(c) It is dimensionally correct but not numerically                                     

(d) It is numerically correct but not dimensionally

Q 16. A B C D Dimensional analysis gives :

         (a) no information about dimensionless constants

         (b) information about dimensionless constants

         (c)   information about dimensionless constants if quantity does not depend upon more than three variables

                (d) information about dimensionless constants if quantity depends upon single variable

Q 17. A B C D A physical quantity depends upon five factors, all of which have dimensions; then method of dimensional analysis:

(a) can be applied                                            
(b) cannot be applied

(c) it depends upon factors involved       
(d) both (a) and (c)

Q 18. A B C D mensional analysis can be used to derive formulae:

(a) containing trigonometrical functions                
(b) containing exponential functions        

(c) containing logarithmic functions                         
(d) none of the above                        

Q 19. A B C D A student when discussing the properties of a medium (except vacuum) writes: velocity of light in vacuum = velocity of light in medium

This formula is:

(a) dimensionally correct              
(b) dimensionally incorrect                  

(c) numerically incorrect                               
(d) both (a) and (c)                           

Q 20. A B C D Given that T stands for time period and l stands for the length of simple pendulum. If g is the acceleration due to gravity, then which of the following statements about the relation T² = (l/8) is correct?

(a) It is correct both dimensionally as well as numerically                            

(b) It is neither dimensionally correct nor numerically

(c) It is dimensionally correct but not numerically                                     

(d) It is numerically correct but not dimensionally                                          

Q 21. A B C D If energy E, velocity V and time T are taken as the fundamental units, the dimensional formula for surface tension is:

(a) EV^-2T^-2            
(b) E^-2VT^-2                            
(c) E^-2V^-2T             
(d) E^-2V^-2T^-2       

Q 22. A B C D If force (F), acceleration (a) and time (T) are used as the fundamental units, the dimensional formula for length will be:

                  (a) F⁰aT²                               
         (b) Fa⁰T²                               
         (c) Fa²T⁰                               
         (d) FaT

Q 23. A B C D If area (A), velocity (v) and density (r) are taken as fundamental units, what is the dimensional formula for force?

(a) Av²r               
(b) A²vr              
(c) Avr²                
(d) Avr               

Q 24. A B C D The liquid drop of density r, radius r and surface tension s oscillates with time period T. Which of the following expressions for T² is correct?

(a) rr³/s              
(b) rs/r³                              
(c) r³s/r              
(d) none of these    

Q 25. A B C D If P represents radiation pressure, c represents speed of light and Q represents radiation energy striking a unit area per second then non-zero integers x, y and z such that P^xQ^yc^z is dimensionless, are :

 (a) x = l, y = l, z = -1         
(b) x = l, y = -l, z = 1          
(c) x = -l, y = l, z = 1          
(d) x = l,  y = l, z = 1

Q 26. A B C D A highly rigid cubical block A of small mass M and side L is fixed rigidly on the other cubical block of same dimensions and of low modulus of rigidity h such that the lower face of A completely covers the upper face of B. The lower face of B is rigidly held on a horizontal surface. A small force F is applied perpendicular to one of the side faces of A. After the force is withdrawn, block A executes small oscillations, the time period of which is given by:                                                                 
  (IIT 1992)

(a)2p $\sqrt{M\eta L}$  

(b)2p$\sqrt{(M\eta /L)}$ $\sqrt{(M\eta /L)}$      

       

(c)2p $\sqrt{M\eta /L}$     

            

(d)$\sqrt{M/\eta L}$ 2p$\sqrt{M/\eta L}$ 

               

Q 27. A B C D Suppose refractive index m is given as                                                                                     

                m = A +  $\frac{B}{{{\lambda }^{2}}}$ 

where A and B are constants and l is wavelength, then dimensions of B are same as that of:

(a) wavelength                 
(b) volume                          
(c) pressure                        
(d) area                

Q 28. A B C D If S is surface tension and R has got units of length, then dimensional formula of S/R is:

                  (a) ML⁰T^-2           
        (b) ML^-1T^-2             
        (c) ML²T^-1             
        (d) ML^-1T^-3         

Q 29. A B C D Height of liquid in a capillary tube is given as 
          h = $\frac{2S\cos \theta }{r\rho g}$ where S is the surface tension of liquid, r is the radius of capillary tube, r is density and g is acceleration due to gravity then dimensional formula for S is :

                 (a) ML⁰T^-2            
        (b) M⁰LT^-2                                          
        (c) ML²T^-2             
        (d) M⁰L⁰T^-3

               

Q 30. A B C D Given that F= -hA$\frac{dv}{dx}$where F is force, A is area and dv/dx is velocity gradient, then dimensional formula of h should be:

(a) ML^-1T^-1           
(b) ML^-1T                              
(c) ML^-2T^-2            
(d) ML²T^-1

Q 31. A B C D The length of a strip measured with a metre rod is 10.0 cm. Its width measured with a vernier callipers is 1.00 cm. The least count of the metre rod is 0.1 cm and that of vernier calipers 0.01 cm. What will be error in its area?

(a) ±0.01 cm²                      
(b) ±0.1 cm²                        
(c) ±0.11 cm²                      
(d) ±0.2 cm²         

Q 32. A B C D The length of a cylinder is measured with a metre rod having least count 0.1 cm. Its diameter is measured with vernier callipers having least count 0.01 cm. Given that length is 5.0 cm and radius is 2.0cm. The percentage error in the calculated value of the volume will be:

(a) 1%                   
(b) 2%                                   
(c) 3%                    
(d) 4%    

Q 33. A B C D The length, breadth and thickness of a block are measured as 125.5 cm, 5.0 cm and 0.32 cm respectively. Which one of the following measurements is most accurate measurement of ... ?

(a) length                            
(b) breadth                         
(c) thickness                       
(d) height             

Q 34. A B C D The percentage errors in the measurement of mass and speed are 2% and 3% respectively. How much will be the maximum error in the estimate of the kinetic energy obtained by measuring mass and speed?

(a) 11%                                 
(b) 8%                                   
(c) 5%                                    
(d) 1%

Q 35. A B C D While measuring the acceleration due to gravity by a simple pendulum, a student makes a positive error of 1% in the length of the pendulum and a negative error of 3% in the value of time period. His percentage error in the measurement of g by the relation g = 4p²(l/T^Z) will be:

(a) 2%                   
(b) 4%                   
(c) 7%                    
(d) 10%               

Q 36. A B C D The best method to reduce random errors is:

(a) to change the instrument used for measurement                             

(b) to take help of experienced observer   

(c) to repeat the experiment many times and to take the average results                 

(d) none of the above                        

Q 37. A B C D The random error in the arithmetic mean of 100 observations is x, then random error in the arithmetic mean of 400 observations would be:

                 (a) 4x                                     
        (b) $\frac{1}{4}$x                                             
        (c) 2x                                     
        (d) $\frac{1}{2}$x

               

Q 38. A B C D Which of the following is the most accurate instrument for measuring length?

(a) Vernier callipers having 20 divisions on the sliding scale which coincide with 19 divisions on the main millimetre scale      

(b) A screw gauge having pitch 1 mm and 50 divisions on the circular scale           

(c)   A vernier scale of least count 0.01 mm 

(d) A screw gauge of least count 0.001 mm

               

Q 39. A B C D The least count of an instrument is 0.01 cm. Taking all precautions the most possible error in the measurement can be:

(a) 0.005 cm                        
(b) 0.001 cm                       
(c) 0.01 cm                          
(d) 0.02 cm          

Q 40. A B C D If DX is the mean absolute error and X_m is the mean value of a physical quantity X, then relative error or fractional error is:

(a) $\overline{\Delta X}$/Xm                       
(b) $\overline{\Delta X}$ + X_m                    
(c) $\overline{\Delta X}$ - X_m                      
(d) X_{m$\overline{\Delta X}$}

Q 41. A B C D The percentage error in above equation will be:

              
        (a) $\frac{\overline{\Delta X}+{{X}_{m}}}{{{X}_{m}}}$´ 100%      
        (b) $\frac{\overline{\Delta X}-{{X}_{m}}}{{{X}_{m}}}$´ 100%    
        (c) $\overline{\Delta X}$/X_m ´ 100%        
        (d) X_m/DX ´ 100%

               

Q 42. A B C D If DX is absolute error in the measurement of X and DY is absolute error in the measurement of Y, then maximum absolute error in the measurement of difference of X and Y is:

(a) ± (DX – DF)                   
(b) ± (DX + DY)                  
(c) ±DX/DF                          
(d) ±DXDF          

Q 43. A B C D [Mean value – Measured value] gives:
        (a) absolute error            
        (b) relative error              
        (c) gross error                    
        (d) random error    

Q 44. A B C D Relative accuracy of a screw gauge can be increased:

(a) by increasing the size of pitch              

(b) by increasing the number of divisions on the circular scale                        

(c) by taking large number of observations 

(d) by having a device free from zero error 

Q 45. A B C D Zero error belongs to the category of:

(a) constant errors          
(b) personal errors          
(c)  instrumental errors 
(d) accidental errors

Q 46. A B C D What is the number of significant figures in 0.310 ´ 10³?

(a) 2                       
(b) 3                       
(c) 4                       
(d) 6                   

Q 47. A B C D What is the number of significant figures in 0.0310?

(a) 2                       
(b) 3                       
(c) 4                       
(d) 5                   

Q 48. A B C D What is the number of significant figures in 200.0?

(a) 1                       
(b) 2                       
(c) 3                       
(d) 4                   

Q 49. A B C D The length of a rod is (11.05 ± 0.05) cm. What is the length of two rods?

(a) (22.1 ± 0.05) cm          
(b) (22.1 ± 0.1) cm            
(c) (22.10 ± 0.05) cm        
(d) (22.10 ± 0.01) cm                           

Q 50.

 A B C D 

 The accuracy of a clock is one part in 10¹⁰. The maximum difference between two such clocks operating for 10¹⁰ seconds is:

(a) 1s                     
(b) 5s                     
(c) 1.0s                  
(d) 10¹⁰ s             

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