41.
The heat generated in a circuit is given by $$Q = {I^2}Rt,$$ where $$I$$ is current, $$R$$ is resistance and $$t$$ is time. If the percentage errors in measuring $$I,R$$ and $$t$$ are $$2\% ,1\% $$ and $$1\% $$ respectively, then the maximum error in measuring heat will be
42.
The values of kinetic energy $$K$$ and potential energy $$U$$ are measured as follows :
$$K = 100.0 \pm 2.0\,J,U = 200.0 \pm 1.0\,J.$$ Then the
percentage error in the measurement of mechanical energy is -
$$\eqalign{
& {M_E} = K + U = 300 \pm 3J = E \pm \Delta E \cr
& \therefore \frac{{\Delta E}}{E} \times 100 = 1\% \cr} $$
43.
A physical quantity $$x$$ depends on quantities $$y$$ and $$z$$ as follows : $$x = Ay + B\tan Cz,$$ where $$A,B$$ and $$C$$ are constants. Which of the following do not have the same dimensions :
$$\left[ x \right] = \left[ A \right] \times \left[ y \right] = \left[ B \right] \Rightarrow \left[ x \right] \ne \left[ A \right]$$
44.
The percentage errors in the measurement of mass and speed are $$2\% $$ and $$3\% $$ respectively. The error in kinetic energy obtained by measuring mass and speed, will be
45.
A thin copper wire of length $$l$$ metre increases in length by $$2\% $$ when heated through $${10^ \circ }C.$$ What is the percentage increase in area when a square copper sheet of length $$l$$ metre is heated through $${10^ \circ }C$$ ?
$$\eqalign{
& W = \vec F \cdot \vec s = F\,s\cos \theta \cr
& = \left[ {ML{T^{ - 2}}} \right]\left[ L \right] = \left[ {M{L^2}{T^{ - 2}}} \right]; \cr
& \vec \tau = \vec r \times \vec F \Rightarrow \tau = r\,F\sin \theta \cr
& = \left[ L \right]\left[ {ML{T^{ - 2}}} \right] = \left[ {M{L^2}{T^{ - 2}}} \right] \cr} $$
48.
If $$p$$ represents radiation pressure, $$c$$ represents speed of light and $$S$$ represents radiation energy striking unit area per $$\sec.$$ The non-zero integers $$x, y, z$$ such that $${p^x}{S^y}{c^z}$$ is dimensionless are
Energy incident per unit area per second $$ = \frac{{{\text{Energy}}}}{{{\text{area}} \times {\text{second}}}} = \frac{{M{L^2}{T^{ - 2}}}}{{{L^2}T}} = M{T^{ - 3}}$$
50.
The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively $$1.5\% $$ and $$1\% $$, the maximum error in determining the density is-