Physical quantities and units：練習題

Which list contains only vector quantities?

A student measures the diameter of a wire using a micrometer screw gauge. The readings are \( 0.22 \text{ mm} \), \( 0.23 \text{ mm} \), \( 0.22 \text{ mm} \), and \( 0.23 \text{ mm} \). The manufacturer states the diameter is \( 0.250 \text{ mm} \).
Which statement best describes the measurements?

Newton's law of gravitation is given by \( F = \frac{G m_1 m_2}{r^2} \), where \( F \) is force, \( m_1 \) and \( m_2 \) are masses, and \( r \) is distance. What are the SI base units of the gravitational constant \( G \)?

Which of the following is a reasonable estimate for the mass of a typical adult human?

What are the SI base units of electrical charge?

A student uses a stopwatch to measure the time for 20 oscillations of a simple pendulum. The student then repeats this process four times to find an average. State whether the timing of 20 oscillations instead of just one reduces random errors or systematic errors, and briefly explain why.

Which of the following lists contains only SI base quantities as defined by the International System of Units?

A ship travels due North at a constant speed of \( 12.0 \text{ m s}^{-1} \) relative to the water. A current in the sea moves due East with a speed of \( 5.0 \text{ m s}^{-1} \). Calculate the magnitude of the resultant velocity of the ship and the angle it makes with the North direction.

A ship travels through the water with two main forces acting on it in the horizontal plane. A tugboat pulls the ship with a force of \( 4500 \text{ N} \) at an angle of \( 30^{\circ} \) North of East. A second tugboat pulls the ship with a force of \( 3000 \text{ N} \) at an angle of \( 45^{\circ} \) South of East.

(a) Distinguish between scalar and vector quantities, providing one example of each from the context of this problem.

(b) By resolving the forces into perpendicular components (North-South and East-West), calculate the magnitude of the resultant force acting on the ship.

The terminal velocity \( v_t \) of a small sphere falling through a fluid is given by \( v_t = \frac{2 r^2 (\rho_s - \rho_f) g}{9 \eta} \), where \( r \) is the radius of the sphere, \( \rho_s \) is the density of the sphere, \( \rho_f \) is the fluid density, \( g \) is the acceleration of free fall, and \( \eta \) is the dynamic viscosity of the fluid.

(a) Show that the equation is homogeneous with respect to SI base units, given that the unit of dynamic viscosity \( \eta \) is \( \text{kg m}^{-1}\text{s}^{-1} \).
(b) An experiment is performed with the following measured values and absolute uncertainties:
Radius \( r = (2.00 \pm 0.02) \text{ mm} \)
Density of sphere \( \rho_s = (7.80 \pm 0.05) \times 10^3 \text{ kg m}^{-3} \)
Fluid density \( \rho_f = (1.20 \pm 0.02) \times 10^3 \text{ kg m}^{-3} \)
Viscosity \( \eta = (0.890 \pm 0.015) \text{ Pa s} \) (note: \( 1 \text{ Pa s} = 1 \text{ kg m}^{-1}\text{s}^{-1} \))
Assuming \( g = 9.81 \text{ m s}^{-2} \) has negligible uncertainty, calculate the terminal velocity \( v_t \) and its percentage uncertainty.