Write dimension of \(\frac{c}{a\times b}\) in relation \(y=aCos\left(\omega t\right)+bt-c\sqrt{t}\). Where \(y\) is displacement, \(t\) is time and \(\omega\) is angular velocity.
The position of the particle at any instant of time t is given by relation \(x\left(t\right)=\left(\frac{v_{o}}{\alpha}\right)\left(1-e^{-\alpha t}\right)\). Where \(v_{o}\) is constant and \(\alpha > 0\). Find: the dimension of \(v_{o}\) and \(\alpha\).
Ans: \([\alpha]=[M^{o}L^{o}T^{-1}]\) and \([v_{o}]=[M^{0}L^{1}T^{-1}]\).
The coefficient of viscosity \(\left(\eta\right)\) of a liquid by the method of flow through a capillary tube is given by the equation, \(\eta = \frac{\pi R^{4}P}{8lV}\), where; R = radius of the capillary tube, l = length of tube, P = Pressure difference between the ends and V = Volume of the liquid flowing through capillary tube per second. Which measurement needs to be made more accurate and why?
A student measure the distance travelled by a body during its free fall. Initially at rest in a given time. He uses this data to estimate the acceleration due to gravity. If, the maximum percentage error in the measurement of the distance and time is \(e_{1}\) and \(e_{2}\) respectively. What will be the percentage error in the estimate of g?
Ans: \(e_{1}+e_{2}\)
A student measures the time period of oscillation of a simple pendulum four times. The data set is 90s, 92s and 95s. If, the minimum division in the measuring clock is \(1s\), then calculate: the mean time reported by the clock?
Ans: \(\left(92\pm2\right)s\).
The following observations were taken for determining the surface tension T of water by capillary tube method. Diameter of capillary tube \(d=\left(1.25\times 10^{-2}\right)m\), rise of water \(h=\left(1.45\times 10^{-2}\right)m\) using \(g=9.8ms^{-2}\) and a simplified equation \(T=\frac{rh\rho}{2}\times 10^{3}Nm^{-1}\). Calculate: the possible error in the surface tension.
Ans: \(1.5\%\).
The diameter of a solid ball is to be determined in an experiment. The diameter of the ball is measured with screw gauge, whose pitch is 0.5 mm and there are 50 divisions on the circular scale. The reading of the main scale is 2.5 mm and that of the circular scale is 20 divisions. If the measurement of mass of the ball has percentage error \(2\%\) then what will be the percentage error in the density?
Ans: \(3.1\%\).
In Searl’s experiment, the diameter of the experimental wire is \(D=0.05 cm\) (Measured by a scale of least count 0.001 cm) and length \(L=110 cm\) (measured by the scale of least count of 0.1 cm). A weight of 50 N causes an extension of l = 0.125 cm (Measured with micrometer of least count of 0.001 cm). Find: the maximum possible error in the value of Young’s Modulus. Screw gauge and metre scale are free from error.
The diameter of a cylinder is measured using vernier callipers with no (zero) error. If is found that zero of the vernier scale lies in between 5.10 cm and 5.15 cm of the main scale. The vernier scale has 50 divisions equivalent to 2.45 cm. The 24th division of the vernier scale exactly coincides with one of the main scale division. Then calculate: the diameter of the cylinder.
