Measurements and Units

A student records the mass of a chemical sample using a digital balance as $0.02050\text{ kg}$. How many significant figures are contained in this measurement?

A simple pendulum on Earth ($g = 10\text{ m/s}^2$) completes $50$ full oscillations in $40\text{ s}$. The pendulum is then transferred to a lunar station where the acceleration due to gravity is $1.6\text{ m/s}^2$, and its length is reduced by $64\%$. What is the time taken, in seconds, for this modified pendulum to complete $30$ oscillations on the lunar station?

A student records the time taken for a simple pendulum to complete $20$ complete oscillations as $30\text{ s}$. What is the period of oscillation of the pendulum, in seconds?

The electrical resistance $R$ of a uniform wire of length $L$ and diameter $d$ is given by $R = \frac{4\rho L}{\pi d^2}$, where $\rho$ is the constant resistivity of the material. If the length $L$ is measured with a percentage error of $2.0\%$ and the diameter $d$ is measured with a percentage error of $1.5\%$, what is the maximum percentage error in the calculated value of $R$?

A ticker-tape timer connected to a power supply operates at a frequency of $50\text{ Hz}$. A continuous strip of paper tape pulled through the timer records a sequence of dots. Calculate the total time interval, in seconds, between the $1\text{st}$ dot and the $21\text{st}$ dot on the tape.

In an experiment to determine the density of a solid sphere, the mass of the sphere is measured as $(50.0 \pm 0.5)\text{ g}$ and its radius is measured as $(1.00 \pm 0.02)\text{ cm}$. What is the maximum percentage error in the calculated density of the sphere?

A student measures the time taken for $40$ complete oscillations of a simple pendulum using a digital stopwatch that has a negative zero error of $-0.40\text{ s}$. If the stopwatch displays a reading of $47.60\text{ s}$ for the oscillations, what will be the correct period of oscillation when the pendulum's length is reduced to one-fourth ($1/4$) of its initial length?

In a physics experiment to determine the acceleration due to gravity $g$ using a free-fall apparatus, the distance of fall is measured as $h = (2.00 \pm 0.04)\text{ m}$ and the duration of fall is measured as $t = (0.50 \pm 0.01)\text{ s}$. Given that $g = \frac{2h}{t^2}$, what is the percentage error in the calculated value of $g$?

In an experiment to determine the Young's modulus $Y$ of a metal wire using the formula $Y = \frac{4FL}{\pi d^2 e}$, the force $F$, length $L$, diameter $d$, and extension $e$ are measured with maximum percentage errors of $1.2\%$, $0.8\%$, $1.5\%$, and $2.0\%$ respectively. What is the maximum percentage error in the calculated value of $Y$?

An experimenter uses a digital stopwatch to record the duration of $40$ complete oscillations of a simple pendulum. The stopwatch displays a reading of $50.8\text{ s}$, but it has a known positive zero error of $+0.8\text{ s}$. What is the correct period of oscillation of the pendulum?

The frequency of a wave is measured as $(200 \pm 4)\text{ Hz}$ and its wavelength is measured as $(0.50 \pm 0.01)\text{ m}$. What is the maximum percentage error in the calculated speed of the wave?

A ticker-tape timer operates at an alternating current frequency of $50\text{ Hz}$. During a mechanics experiment, a paper tape pulled through the device records a section containing $11$ consecutive dots. What is the total time elapsed for this section of the tape?

The fundamental frequency $f$ of a stretched vibrating string depends on the tension force $F$, the length of the string $l$, and its linear mass density $\mu$ (mass per unit length) according to the relation $f = k F^a l^b \mu^c$, where $k$ is a dimensionless constant. Which of the following sets of exponents $(a, b, c)$ correctly satisfies dimensional homogeneity?

Match each physical quantity listed on the left with its correct classification and SI unit definition on the right.

Match each derived physical quantity in Column I with its equivalent SI unit expressed in terms of fundamental (base) units in Column II.

A micrometer screw gauge has a pitch of $0.5\text{ mm}$ and $50$ divisions on its circular thimble scale. When its anvil and spindle are fully brought together without any object, the zero mark of the thimble scale lies $4$ divisions past the main scale index line in the tightening direction (a positive zero error). When the wall thickness of a hollow metal pipe is measured with this micrometer, the main scale reading is $3.5\text{ mm}$ and the $32\text{nd}$ thimble division aligns with the index line. If the outer diameter of the pipe is separately measured to be $25.40\text{ mm}$, what is the corrected inner diameter of the metal pipe in millimeters?

Which of the following combinations of fundamental SI base units is equivalent to the volt ($\text{V}$), the SI unit of electric potential difference?

A rocket carries a payload of mass $25\text{ kg}$ resting on a spring balance. During its vertical ascent, the spring balance indicates a reading of $350\text{ N}$. Taking the acceleration due to gravity as $g = 10\text{ m s}^{-2}$, what is the magnitude of the upward acceleration of the rocket in $\text{m s}^{-2}$?

The derived SI unit of electric capacitance, the farad ($\text{F}$), can be expressed in terms of fundamental SI base units as $\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d$. What is the numerical value of the sum of the exponents $a + b + c + d$?

A spring balance calibrated in newtons shows an initial pointer reading of $+0.4\text{ N}$ before any load is suspended. When a stone is attached to the balance hook, the pointer indicates $18.4\text{ N}$. Taking acceleration due to gravity $g = 10\text{ m s}^{-2}$, what is the actual mass of the stone?