Units & Dimensions, Measurements, Vectors

1. The time dependence of a physical quantity P is given by where α is a constant and t is time. Then constant α is

Ans:  has dimension of T^-2
Hint:  Here αt² is dimensionless.   and has dimensions of [T^-2]

2. Write dimensions of   in the relation where P is pressure, x is distance, t is the time.
Ans:  M^-1 L⁰ T^-2
Hint: Here 'a' has dimensions of  't' as [T² ]  

3. If L, R, C and V respectively represent inductance, resistance, capacitance and potential L difference then dimensions of    are the same as those of

Ans:   

Hint:
          

              
4. Of the following quantities which one has dimensions different from the remaining three
1) Energy density
2) Force per unit area
3) Angular momentum per unit mass
4) Product of charge per unit volume and voltage.
Ans: Angular momentum per unit mass

Hint:

 
     Product of charge per unit volume and
       

5. The frequency f of vibrations of a mass 'm' suspended from a spring of force constant 'k' is given by f = cm^x k^y where c is dimensionless constant. The value of x and y are respectively.

Ans:  ^:  
_{Hint:    f = cm^x × k^y}
         
         
 

6. Given R₁ = 5.0 ± 0.2 Ω, R₂ = 10.0 ±0.1 Ω. What is total resistance in parallel with possible percentage error?
Ans:  3.3 Ω ± 7 %

Hint:

7. The % error in the measurement of the length of a simple pendulum and its time period are 2% and 3% respectively. The maximum error in the value of acceleration due to gravity obtained from those measurements is
Ans:  8%

Hint:  

               log g = log 4π² + log l − 2 log T
               maximum error in 'g' is
               

               
8. The mass of a body is 20.000 g and its volume is 10.00 cm³. If the measured values are expressed to the correct significant figures, the maximum error in the value of density is 

Ans: 0.01  cm³  
Hint: Maximum error in measuring mass  = 0.001 gm (.^.. LC = 0.001g)
          Maximum error in measuring volume is = 0.01 cm³

9. There are four different expressions for displacement 'y' in a periodic motion as a function of time 't', 'a' as amplitude, "T' as time period. Which of the following can be correct?

1)
_{2) y = a sin V t
³⁾}

4)

Hint: Verify given equations in such a way that LHS = RHS

10. A physical quantity x is calculated from the relation . If percentage error in a, b, c and d are 2%, 1%, 3%, 4% respectively. What is the % error in 'x'?
Ans:   12%

Hint: 

             = 4 + 3 + 3 + 2 = 12%
11. Using significant figures match the following

Ans:  4-a, 3-b, 2-d, 1-c

12. 

Ans:  64

Hint:  
    
 

13. Unit vectors  and  are inclined at an angle θ, then 
Ans:   
Hint:   

14. The resultant of two vectors P and Q is R. If the magnitude of Q is doubled. The new resultant vector becomes perpendicular to P. Then the magnitude of R is equal to

Ans: Q
Hint: :
      
               R² = P² + Q² + 2PQ cosθ
            

15. What is the resultant of three coplanar forces: 300 N at 0^o, 400 N at 30°, and 400 N at 150° ?

Ans:  500 N
Hint:  Net force along X- axis is
           F_x= F₁ + F₂ cos 30^o - F₃ cos 30^o

                
                
          F_y = F₂ sin 30° + F₃ sin 30°

16. Given that A + B + C = 0 out of three vectors two vectors are equal in magnitude and magnitude of third vector is  times that of either of two having equal magnitude. The angles between vectors are
Ans:  90°^o, 135°^o, 135°^o

Hint:    

           Apply Lami's theorem
           
                      β = 45°      α = β
                      γ = 180 - 2α = 90°,
                      Angle between  = 180 - γ = 90°
                      Angle between  = 180 - α = 135°
                      Angle between  = 180 - β = 135°