--- title: "Crank and slotted lever velocity analysis" subject: "Theory of Mechanism and Machine I" chapter: 7 key: "semester_6/theory_of_mechanism_and_machine_i:crank-slotted-lever-velocity-analysis" source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/crank-slotted-lever-velocity-analysis site: BCT Engineering --- # Crank and slotted lever velocity analysis Theory of Mechanism and Machine I · Chapter 7: Kinematic Analysis of Mechanisms - Asked in: year not given, 6 marks - Video: [Velocity Analysis in Crank and Slotted Lever Mechanism | Online Video-Tutorial | Theory of Machines](https://www.youtube.com/watch?v=FlPsFS_maEQ) — Magic Marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/crank-slotted-lever-velocity-analysis - Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-7 ## Question In a crank and slotted-lever quick return mechanism, the crank pivot O1 and the lever pivot O2 are 300 mm apart on a horizontal line, with O2 to the right of O1. The crank O1B is 120 mm long and rotates anticlockwise at 10 rad/s. A sliding block at B moves in the slot of the lever pivoted at O2. At the instant when the crank is vertical (B directly above O1), find the angular velocity of the slotted lever, the velocity of sliding of the block in the slot and the velocity of the point D on the lever, 500 mm from O2 on the side of O2 away from B. ## Short answer Resolve crank pin velocity into slot-parallel and perpendicular components. Angular velocity of slotted lever is 1.379 rad/s anticlockwise, sliding velocity 1.114 m/s, velocity of point D 0.690 m/s. ## Answer #### Vector equation Let B$_2$ be the crank pin (block) and B$_3$ the coincident point on the lever. The block slides along the lever, so $$ \vec v_{B_2} = \vec v_{B_3} + \vec v_{B_2B_3} $$ - $v_{B_2}$: magnitude $\omega\cdot O_1B = 10 \times 0.12 = 1.20$ m/s, perpendicular to $O_1B$ (horizontal, towards the left, since the crank is vertical and rotates anticlockwise). - $v_{B_3}$: perpendicular to $O_2B$ (unknown magnitude). - $v_{B_2B_3}$: along the slot $O_2B$ (unknown magnitude). #### Geometry $O_2B = \sqrt{300^2 + 120^2} = 323.1$ mm. The slot makes an angle $\alpha = \tan^{-1}(120/300) = 21.80^\circ$ with the horizontal, so $\cos\alpha = 0.9285$ and $\sin\alpha = 0.3714$. #### Velocity polygon (resolving $v_{B_2}$) ```text b2 <---- o o: pole \ ' | along \ ' | perp. to O2B slot \ | b3 ``` Draw $o b_2$ = 1.2 m/s horizontal. Through $o$ draw a line perpendicular to $O_2B$ (direction of $v_{B_3}$), and through $b_2$ draw a line parallel to $O_2B$ (direction of $v_{B_2B_3}$). They meet at $b_3$. Resolving $v_{B_2}$ in these two directions: $$ v_{B_2B_3} = v_{B_2}\cos\alpha = 1.2 \times 0.9285 = 1.114\ \text{m/s} $$ $$ v_{B_3} = v_{B_2}\sin\alpha = 1.2 \times 0.3714 = 0.446\ \text{m/s} $$ #### Angular velocity of lever $$ \omega_{lever} = \frac{v_{B_3}}{O_2B} = \frac{0.446}{323.1/1000} = 1.379\ \text{rad/s (anticlockwise)} $$ Sense: B$_3$ lies up and to the left of $O_2$, and its velocity (perpendicular to the lever) points down and to the left, so the lever turns anticlockwise about $O_2$. #### Velocity of D $$ v_D = \omega_{lever}\cdot O_2D = 1.379 \times 0.5 = 0.690\ \text{m/s} $$ perpendicular to the lever (D is on the opposite side of $O_2$, so it moves in the opposite direction to $B_3$). **Answer: $\omega_{lever} = 1.379$ rad/s anticlockwise, sliding velocity $= 1.114$ m/s, $v_D = 0.690$ m/s.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the vector equation used for the block at B? A. v_B2 = v_B3 + v_B2B3 (sliding along the slot) B. v_B2 = v_O2 + v_B3 C. v_B2 = v_B3 x v_B2B3 D. v_B3 = v_B2 + omega Answer: A. The block B2 has the velocity of the coincident lever point B3 plus its sliding velocity along the slot. (Key point: Vector equation) 2. What are the directions of v_B3 and v_B2B3? A. Both perpendicular to O2B B. v_B3 along O2B and v_B2B3 perpendicular C. v_B3 perpendicular to O2B and v_B2B3 along the slot O2B D. Both horizontal Answer: C. B3 moves with the lever, so perpendicular to O2B; sliding is along the slot. (Key point: Directions in polygon) 3. Given v_B2 = 1.2 m/s and slot angle alpha = 21.8 degrees, the sliding velocity is: A. 0.446 m/s B. 1.2 m/s C. 0.690 m/s D. 1.114 m/s Answer: D. v_B2B3 = v_B2 cos(alpha) = 1.2 x 0.9285 = 1.114 m/s. (Key point: Sliding velocity) 4. How is the lever's angular velocity found? A. omega = v_B2/O1B B. omega = v_B3/O2B = 0.446/0.3231 = 1.379 rad/s C. omega = v_B2B3/O2B D. omega = v_D/O2B Answer: B. The lever rotates about O2, so omega = v_B3/O2B = 1.379 rad/s. (Key point: Lever angular velocity) ## Related in Theory of Mechanism and Machine I - [Slider crank velocity by instantaneous centre method](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aslider-crank-velocity-instantaneous-centre) — Theory of Mechanism and Machine I, chapter 7 - [Position analysis of a four-bar mechanism](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Afour-bar-position-analysis-numerical) — Theory of Mechanism and Machine I, chapter 2 - [Static force analysis of slider crank](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Astatic-force-analysis-slider-crank) — Theory of Mechanism and Machine I, chapter 8 - [Four-bar angular velocities by IC method](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Afour-bar-velocity-by-ic-method) — Theory of Mechanism and Machine I, chapter 7 - [Inversions of single slider crank chain](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aslider-crank-chain-four-inversions) — Theory of Mechanism and Machine I, chapter 1 ## Related in other subjects - [Velocity analysis of slider-crank mechanism](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Acrank-slider-velocity-numerical) — Engineering Mechanics, chapter 5 - [Shaper types and quick return mechanism](https://www.bctengineer.com/question.md?key=semester_2%2Fworkshop_technology%3Ashaper-types-quick-return) — Workshop Technology, chapter 6 - [Crank AB 5 rad/s, piston velocity](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Acrank-5rads-piston-velocity-coriolis) — Applied Mechanics (Dynamics), chapter 4 - [Acceleration analysis of slider-crank](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Acrank-slider-acceleration-numerical) — Engineering Mechanics, chapter 5