Download 10800 DNK201E Dynamics

12366 dnk201e dynamics - sample problems 3
Problem 1: (15p) The tennis ball is struck with a
horizontal velocity 𝑣⃗𝐴 , strikes the smooth ground at B, and
bounces upward at 𝜃 = 30°. Determine the initial velocity
𝑣⃗𝐴 , the final velocity 𝑣⃗𝐵 , and the coefficient of restitution
between the ball and the ground.
2.28 m
6m
Problem 2: (25p) In the speed-governing mechanism 𝑚 = 0.2𝑘𝑔, 𝐿 =
10 𝑐𝑚 (rods AB have negligible mass), 𝜔 = 500 𝑟𝑝𝑚, 𝜃 = 40°, 𝜃̇ =
0. Determine 𝜃̈ at that instant and the tension in the rod.
0.3 m 0.3 m
C
B
Problem 3: (30p) If link CD has an angular velocity of
𝜔𝐶𝐷 = 6 𝑟𝑎𝑑/𝑠,
E
(a) determine the velocity of point B,
(b) and determine the acceleration of point B.
0.6 m
A
30°
Problem 4: (30p) A 27 𝑘𝑔 ladder is assumed to be a uniform bar which
starts sliding without frictional resistance. Determine the forces at A and B.
(𝐼𝐴 = 𝐼𝐵 = 𝑚𝑙 2⁄3 , 𝐼𝐶 = 𝑚𝑙 2 ⁄12)
𝜔𝐶𝐷 = 6 𝑟𝑎𝑑/𝑠
5m
D
Kinematics and Kinetics of a Particle:
Coordinates Position (𝑟⃗)
Rectangular
⃗⃗
𝑥𝑖⃗ + 𝑦𝑗⃗ + 𝑧𝑘
Polar
𝑟𝑛⃗⃗𝑟
Cylindrical
⃗⃗
𝑟𝑛⃗⃗𝑟 + 𝑧𝑘
Spherical
𝑟𝑛⃗⃗𝑟
Velocity (𝑣⃗)
⃗⃗
𝑥̇ 𝑖⃗ + 𝑦̇ 𝑗⃗ + 𝑧̇ 𝑘
̇
𝑟̇ 𝑛⃗⃗𝑟 + 𝑟𝜃𝑛⃗⃗𝜃
⃗⃗
𝑟̇ 𝑛⃗⃗𝑟 + 𝑟𝜃̇𝑛⃗⃗𝜃 + 𝑧̇ 𝑘
𝑟̇ 𝑛⃗⃗𝑟 + 𝑟𝜙̇𝑛⃗⃗𝜙
+ 𝑟𝜃̇ sin 𝜙 𝑛⃗⃗𝜃
Acceleration (𝑎⃗)
⃗⃗
𝑥̈ 𝑖⃗ + 𝑦̈ 𝑗⃗ + 𝑧̈ 𝑘
2
̇
(𝑟̈ − 𝑟𝜃 )𝑛⃗⃗𝑟 + (𝑟𝜃̈ + 2𝑟̇ 𝜃̇)𝑛⃗⃗𝜃
⃗⃗
(𝑟̈ − 𝑟𝜃̇ 2 )𝑛⃗⃗𝑟 + (𝑟𝜃̈ + 2𝑟̇ 𝜃̇)𝑛⃗⃗𝜃 + 𝑧̈ 𝑘
(𝑟̈ − 𝑟𝜙̇ 2 − 𝑟𝜃̇ 2 sin2 𝜙)𝑛⃗⃗𝑟
+(𝑟𝜙̈ + 2𝑟̇ 𝜙̇ − 𝑟𝜃̇ 2 sin 𝜙 cos 𝜙)𝑛⃗⃗𝜙
+(𝑟𝜃̈ sin 𝜙 + 2𝑟̇ 𝜃̇ sin 𝜙 + 2𝑟𝜙̇𝜃̇ cos 𝜙)𝑛⃗⃗𝜃
Tangential and normal components: 𝑣⃗ = 𝑣𝑛⃗⃗𝑡 , 𝑎⃗ = 𝑣̇ 𝑛⃗⃗𝑡 + (𝑣 2 ⁄𝜌)𝑛⃗⃗𝑛
3/2
Radius of curvature: 𝜌 =
[1+(𝑑𝑦⁄𝑑𝑥 )2 ]
𝑑2 𝑦/𝑑𝑥 2
Work-energy principle: 𝑈12 = 𝑇2 − 𝑇1
2
Work of a force: 𝑈12 = ∫1 𝐹⃗ ∙ 𝑑𝑟⃗
𝑑𝑈
Power: = 𝐹⃗ ∙ 𝑣⃗
𝑑𝑡
⃗⃗V, Conservation of mechanical energy: 𝑇1 + 𝑉1 = 𝑇2 + 𝑉2
Conservative force: 𝐹⃗ = −∇
𝑑
Time rate of change of linear momentum: ∑ 𝐹⃗ = 𝐺⃗̇ = 𝑑𝑡 (𝑚𝑣⃗)
⃗⃗̇0 = 𝑑 (𝑟⃗ × 𝑚𝑣⃗) = 𝑟⃗ × 𝐹⃗ , ∑ 𝑀
⃗⃗⃗0 = 𝐻
⃗⃗̇0
Time rate of change of angular momentum: 𝐻
𝑑𝑡
∑ 𝑀𝑥 = 𝐻̇𝑥 ∑ 𝑀𝑦 = 𝐻̇𝑦 ∑ 𝑀𝑧 = 𝐻̇𝑧
Angular impulse:
𝑡
⃗⃗⃗𝑖 𝑑𝑡 = 𝐻
⃗⃗𝑂 − 𝐻
⃗⃗𝑂 = ∑𝑛𝑖=1(𝑟⃗ × 𝑚𝑖 𝑣⃗𝑖 )2 − ∑𝑛𝑖=1(𝑟⃗ × 𝑚𝑖 𝑣⃗𝑖 )1
∑𝑛𝑖=1 ∫𝑡 2 𝑀
𝑂
2
1
1
Kinematics of Rigid Bodies:
Velocities in general plane motion: 𝑣⃗𝐵 = 𝑣⃗𝐴 + 𝜔
⃗⃗ × 𝑟⃗𝐵/𝐴
Accelerations in general plane motion: 𝑎⃗𝐵 = 𝑎⃗𝐴 + 𝛼⃗ × 𝑟⃗𝐵/𝐴 + 𝜔
⃗⃗ × (𝜔
⃗⃗ × 𝑟⃗𝐵/𝐴 )
Kinetics of Rigid Bodies in Plane Motion:
∑ 𝐹𝑥 = 𝑚𝑎𝐶,𝑥
∑ 𝐹𝑦 = 𝑚𝑎𝐶,𝑦
∑ 𝑀𝐶 = 𝐼𝐶 𝛼