113 年 國立成功大學系統及船舶機電工程研究所乙組《動力學》

(1) See Fig. 1. If the car decelerates uniformly along the curved road from 23 m/s at A to 15 m/s at C, determine the velocity $v_B$ [10%] and acceleration $a_B$ [10%] of the car at B.

🖼️【此處有附圖，請對照原卷】
Fig. 1

Given:
Initial velocity at A, $v_A = 23$ m/s
Final velocity at C, $v_C = 15$ m/s
The deceleration is uniform along the curved path.
Radius of curvature at B, $\rho_B = 300$ m
Distance from A to C along the curve is $250 + 50 = 300$ m.

Determine:
Velocity at B, $v_B$
Acceleration at B, $a_B$

(2) The drinking fountain is designed such that the nozzle is located from the edge of the basin as shown in Fig. 2. Determine the maximum and minimum speed: $(v_A)_{\text{max}}$ [10%] and $(v_A)_{\text{min}}$ [10%] at which water can be ejected from the nozzle so that it does not splash over the sides of the basin at B and C.

🖼️【此處有附圖，請對照原卷】
Fig. 2

Given:
The nozzle is at point A.
Point B is at an angle of $40^\circ$ from the horizontal, at a horizontal distance of $250$ mm and a vertical distance of $53$ mm from A.
Point C is at a horizontal distance of $250$ mm from A, and at the same vertical level as A.
Assume $g = 9.81$ m/s$^2$.

Determine:
Maximum initial speed $(v_A)_{\text{max}}$
Minimum initial speed $(v_A)_{\text{min}}$
such that water does not splash over B and C.

Note: $250 \text{ mm} = 0.25 \text{ m}$, $53 \text{ mm} = 0.053 \text{ m}$.

(3) See Fig. 3. Two disks A and B each have a mass of 1 kg and the initial velocities shown just before they collide (subscript 1). If the coefficient of restitution is $e = 0.66$, determine the x- and y-component of the velocity of disk A and B just after impact (subscript 2): $(v_A)_{2x}$=? [5%] $(v_A)_{2y}$=? [5%] $(v_B)_{2x}$=? [5%] $(v_B)_{2y}$=? [5%]

🖼️【此處有附圖，請對照原卷】
Fig. 3

Given:
Mass of disk A, $m_A = 1$ kg
Mass of disk B, $m_B = 1$ kg
Initial velocity of disk A, $(v_A)_1 = 4$ m/s in the y-direction.
Initial velocity of disk B, $(v_B)_1 = 3$ m/s in the direction making an angle with the x-axis.
Coefficient of restitution, $e = 0.66$.
The plane of contact is the y-axis.

Determine:
$(v_A)_{2x}$, $(v_A)_{2y}$
$(v_B)_{2x}$, $(v_B)_{2y}$

Note: From Fig. 3, the initial velocity of disk B $(v_B)_1$ is in the direction specified by the vector $(4, 3)$.
The magnitude of $(v_B)_1$ is $\sqrt{4^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5$ m/s.
The direction vector for $(v_B)_1$ is $\mathbf{u}_B = \frac{4\mathbf{i} + 3\mathbf{j}}{5}$.
So, $(v_B)_{1x} = 5 \cdot \frac{4}{5} = 4$ m/s and $(v_B)_{1y} = 5 \cdot \frac{3}{5} = 3$ m/s.

Initial velocities:
$(v_A)_1 = 0 \mathbf{i} + 4 \mathbf{j}$ m/s
$(v_B)_1 = 4 \mathbf{i} + 3 \mathbf{j}$ m/s

(4) As shown in Fig. 4, block A and B each have a mass 5-kg. For $\theta = 42^\circ$, determine the largest horizontal force $P$ [10%] which can be applied to B so that it will not slide on A. Also, what is the corresponding acceleration $a$? [10%] The coefficient of static friction between A and B is $\mu_s = 0.3$. Neglect any friction between A and the horizontal surface.

🖼️【此處有附圖，請對照原卷】
Fig. 4

Given:
Mass of block A, $m_A = 5$ kg
Mass of block B, $m_B = 5$ kg
Angle, $\theta = 42^\circ$
Coefficient of static friction between A and B, $\mu_s = 0.3$.
Horizontal force $P$ is applied to block B.
Friction between A and the horizontal surface is neglected.

Determine:
Largest horizontal force $P$ such that block B does not slide on block A.
Corresponding acceleration $a$ of the system (A and B move together).