Scalars and vectors：練習問題

A force of magnitude \( 50.0\, \text{N} \) acts on a particle at an angle of \( 30.0^{\circ} \) to the vertical. What is the magnitude of the horizontal component of this force?

Which pair of quantities consists of a scalar and a vector, respectively?

Vector \(\mathbf{A}\) has a magnitude of \(10.0\, \text{units}\) acting North. Vector \(\mathbf{B}\) has a magnitude of \(5.0\, \text{units}\) acting West. Calculate the magnitude of the resulting vector \((\mathbf{A} - 2\mathbf{B})\).

A man pulls a trolley with a force of \(150\, \text{N}\) at an angle of \(30^\circ\) above the horizontal. What is the magnitude of the horizontal component of the force?

A swimmer intends to cross a river of width \(60\, \text{m}\) where the current flows at a constant speed of \(1.2\, \text{m s}^{-1}\). The swimmer's speed in still water is \(2.0\, \text{m s}^{-1}\). If they aim their body at an angle such that their resultant path is exactly perpendicular to the river banks, calculate the time taken to cross the river.

A boat attempts to travel directly across a river flowing due East at \(3.0\text{ m s}^{-1}\). The boat's engine provides a velocity of \(4.0\text{ m s}^{-1}\) relative to the water, directed due North. Calculate the magnitude of the resultant velocity of the boat relative to the bank.

A swimmer heads North across a lake at \(1.5\text{ m s}^{-1}\). If the lake current is flowing East at \(0.50\text{ m s}^{-1}\), calculate the angle the resultant velocity makes with the North direction.

Explain why displacement is a vector quantity, while distance is a scalar quantity, and give one physical scenario where the magnitude of displacement is zero but the distance traveled is non-zero.

A ship sails due East with a constant velocity of magnitude \(v_S = 15.0\) km h\(^{-1}\). At the same time, the wind exerts a force on the ship resulting in an additional velocity component due North of magnitude \(v_W = 8.0\) km h\(^{-1}\).

(a) Define a scalar quantity and a vector quantity, giving one example of each from the context of this problem (excluding velocity).
(b) Calculate the magnitude of the resultant velocity of the ship.
(c) Determine the direction of the resultant velocity, measured clockwise from North.

Three coplanar forces, \(F_1\), \(F_2\), and \(F_3\), act at a single point O and keep the object in equilibrium.

The forces are:
\(F_1 = 12.0\) N acting due East.
\(F_2 = 18.0\) N acting North \(40.0^{\circ}\) West (i.e., \(40.0^{\circ}\) measured away from North towards West).
\(F_3\) is the third force necessary for equilibrium.

(a) Resolve \(F_1\) and \(F_2\) into their horizontal (East) and vertical (North) components.
(b) Use the conditions for equilibrium to determine the magnitude of the horizontal and vertical components of \(F_3\).
(c) Calculate the magnitude of force \(F_3\).
(d) Determine the direction of force \(F_3\), expressed as an angle measured clockwise from the positive East direction.