Measurements and Units

The coefficient of dynamic viscosity $\eta$ of a fluid can be defined using Newton's formula for viscous flow, $\eta = \frac{F}{A \frac{\Delta v}{\Delta x}}$, where $F$ is the viscous drag force, $A$ is the contact surface area, and $\frac{\Delta v}{\Delta x}$ is the velocity gradient perpendicular to the flow. Which of the following combinations of fundamental SI base units is equivalent to the coefficient of dynamic viscosity?

The dynamic pressure $P$ exerted by a moving fluid depends on its density $\rho$ and flow velocity $v$ according to the relationship $P = k \rho^a v^b$, where $k$ is a dimensionless constant. What is the value of the exponent $b$?

Newton's law of universal gravitation states that the gravitational force $F$ between two point masses $m_1$ and $m_2$ separated by a distance $r$ is given by $F = \frac{G m_1 m_2}{r^2}$, where $G$ is the universal gravitational constant. What is the dimensional formula of $G$?

A Vernier caliper with a least count of $0.01\text{ cm}$ is used to measure the internal diameter of a cylindrical pipe. When the jaws are fully closed without any object between them, the zero mark of the Vernier scale lies to the right of the zero mark on the main scale, and the $3\text{rd}$ Vernier division coincides with a main scale division. During measurement, the main scale reads $2.40\text{ cm}$ and the $6\text{th}$ Vernier division coincides with a main scale mark. What is the corrected internal diameter of the pipe?

The force constant (stiffness) $k$ of a helical spring measures its resistance to elastic deformation and is defined by the relationship $k = \frac{F}{e}$, where $F$ represents the restoring force and $e$ represents the extension. Which of the following represents the SI unit of $k$ expressed strictly in terms of fundamental SI base units?

A chemical balance is used to determine the mass of a metallic sphere, giving a reading of $6.0\text{ kg}$. The sphere is then suspended from a spring balance at a laboratory where the local acceleration due to gravity is $10\text{ m s}^{-2}$. If the spring balance reads $-3.0\text{ N}$ before any load is attached, what is the indicated reading on the spring balance when the sphere is suspended?

A body is weighed on a distant planet where the acceleration due to gravity is $4\text{ m s}^{-2}$. A spring balance calibrated on Earth ($g = 10\text{ m s}^{-2}$) gives a reading of $20\text{ N}$ for the body on this planet. What is the mass of the body as measured by a beam balance on the planet?

A Vernier caliper has $20$ divisions on its Vernier scale that coincide with $19$ main scale divisions of $1\text{ mm}$ each. When the jaws of the instrument are brought together without any object, the zero of the Vernier scale lies to the right of the main scale zero mark, and the $3\text{rd}$ Vernier division coincides with a main scale mark. When used to measure the thickness of a wooden block, the main scale reads $3.5\text{ cm}$ and the $12\text{th}$ Vernier division coincides with a main scale line. What is the actual thickness of the wooden block?

A spring balance is calibrated in kilograms at sea level where $g = 10.0\text{ m s}^{-2}$. An object measured with an equal-arm beam balance at sea level is found to have a mass of $36.0\text{ kg}$. If this object is taken to a high altitude where $g = 9.0\text{ m s}^{-2}$ and suspended from the same spring balance, what reading will the scale of the spring balance display?

A spring balance suspended inside an elevator reads $48\text{ N}$ when supporting an object while the elevator accelerates downwards at $2\text{ m s}^{-2}$. Taking acceleration due to gravity $g = 10\text{ m s}^{-2}$, what is the mass of the object as measured by an equal-arm beam balance?

Match each length measuring instrument on the left with its appropriate application and precision on the right.

Match each physical quantity or measuring instrument related to mass and weight in Column A with its corresponding physical property or operational principle in Column B.