Constrained Motion

CONSTRAINED MOTION: PULLEY BLOCK SYSTEM & APPLICATION OF STRING CONSTRAINTS Introduction to Constrained Motion When an object is compelled to travel within defined limitations. In this context, the object is constrained to travel along the surface of the block due to applied force and subsequently descends vertically due to the influence of gravity. [Image showing a block on a surface with a force F applied to it and gravity g acting downwards.] Similarly, in this case, the block is restricted to slide along an inclined plane. The force responsible for imposing this constraint on the body is referred to as the constraint force. [Image showing a block on an inclined plane with gravity g acting on it.] There are two primary constraints: String/Rod constraints Wedge constraints String Constraints Consider two objects linked by a string possessing the following characteristics: The string maintains a constant length, signifying that it is inextensible. This implies that if one block moves a distance of x, the other block also moves the same distance. This relationship can be expressed mathematically as: X_A = X_B Where X_A is displacement of block A and X_B is displacement of block B. [Image showing two blocks, A and B, connected by a string on a flat surface. Block A has moved a distance X_A and block B has moved a distance X_B.] [Image of a pulley system with two blocks, A and B. Block A has a displacement X_A and block B has a displacement X_B.] It consistently remains taut and does not become slack. The motion characteristics of these objects along the length of the string and in the direction of extension possess a specific relationship among their parameters. In this context, the parameters of motion refer to displacement, velocity, and acceleration, and the objects denote the bodies directly connected to the string. A definite relationship refers to the constraint relation, such as X_A = X_B. Constraint Relations Mathematical relationships that describe the motion of two or more bodies within a system are referred to as constraint relations. [Image showing two blocks, A and B, connected by a string. A force F is pulling on the string attached to block B. The velocities are labeled V_A and V_B, and the displacements X_A and X_B.] In the provided system, given that the length of the string remains constant. X_A = X_B Taking the derivative of the equation with respect to time. dX_A/dt = dX_B/dt It is known that dX_A/dt = V_A and dX_B/dt = V_B V_A = V_B Taking the second derivative of the equation with respect to time. dV_A/dt = dV_B/dt However, it is known that dV_A/dt = a_A and dV_B/dt = a_B a_A = a_B Hence, we can establish a connection between the parameters by utilizing the inextensibility constraint of the string. Therefore, the constraint relations in this case are X_A = X_B, V_A = V_B and a_A = a_B Pulley-Block System Atwood’s machine Atwood's machine refers to a contraption where two objects are linked by a string passing over a pulley in such a way that if one descends, the other ascends. In this setup, heavier objects will descend, while lighter objects will ascend. [Image of a simple Atwood's machine with two masses, m1 and m2, connected by a string over a single pulley.] Examine the magnitude of displacement for two objects. According to the inextensibility constraint of the string, x₁ = x₂ Likewise, since they are covering an equal distance within the same time interval, v₁ = v₂ a₁ = a₂ Example In the case of the depicted blocks, the string is inextensible. Examine the magnitude of displacement for the two objects, x₁ = x₂ Similarly, since they are traversing an equal distance within the same time interval, v₁ = v₂ a₁ = a₂ Ex. Determine the connection between the velocities of blocks A and B. [Diagram of a pulley system with two blocks, A and B. Block A is attached to a movable pulley, and the string passes over a fixed pulley to block B.] Sol. In the provided system, there are two movable pulleys and two mobile blocks. Adopt the depicted velocities and variables as indicated in the figure. l₁ + l₂ + l₃ = 0 (-v_A cos θ) + 0 + v_p = 0 v_A cos θ = v_p ... (1) Likewise, for the second string. l₃ + l₄ = 0 +(v_p - v_B) + v_p = 0 v_p = 2v_B ... (2) From (1) and (2) v_A cos θ = 2(2v_B) v_A cos θ = 4v_B [A more detailed diagram of the same pulley system, with lengths l1, l2, l3, l4 and velocities v_A, v_B, v_p labeled.] Ex. Determine the constraint relation between points A, B, and C as a ring slides to the right on a rod. [Diagram of a system where a ring A slides on a horizontal rod. A string is attached to the ring, passes through a series of pulleys, and connects to blocks B and C.] Sol. As the ring slides, the movable pulley descends. Subsequently, based on the velocity components of the endpoints along the string's length, v_A cos θ = v_p For the movable pulley, applying the formula derived for pulley-block interaction, v_p = (-v_B - v_C)/2 Since both the pulley and the two blocks are descending, substitute the values of the velocity variables for the pulley, block C, and block B. v_A cos θ = (v_B + v_C)/2 [Image showing a pulley system with a ring A on a rod and two blocks B and C. Various velocity components are labeled, such as v_A, v_p, v_B, v_C, and v_A cos θ.] v_p = v_A cos θ [Diagram showing two different pulley systems connecting masses m1 and m2.] Hence, we can represent the system in a mathematical equation form. Ex. System with movable pulley Recognize all the elements and the number of strings involved in the problem. In this scenario, the distance traversed by the movable pulley is identical to the distance covered by block 2 since they are connected by a single inextensible string. While the movable pulley descends by the distance x₂, the strings on both sides of the pulley must also move downward by the same distance, namely x₂. Therefore, lowering the movable pulley by the distance x₂ involves a length of string of the size 2x₂. This length is entirely provided by the left side of the string, as the total length remains constant. [Diagram showing a system with a fixed pulley and a movable pulley, connecting masses m1 and m2.] This establishes a correlation between x₁ and x₂. Thus, x₁ = 2x₂ v₁ = 2v₂ And a₁ = 2a₂ Steps for writing string constraint relation Step 1: Recognize all the elements and count the number of strings in the given scenario. The depicted system comprises two objects and two strings, as illustrated in the figure. Step 2: Establish variables and orientations to symbolize the motion parameters, including displacement, velocity, acceleration, etc. In the system, the two blocks, string, and a pulley exhibit mobility. Assign variables x₁ and v₁ to denote the velocities of block 1 and 2, respectively, and designate v_p as the velocity of the pulley. Step 3: Recognize a string, partition it into distinct linear segments, and formulate the constraint equation. In this context, the upper string is segmented into lengths l₁, l₂, and l₃, while the lower string has a length of l₄, as depicted in the figure. Given that the total length of any inextensible string always remains constant, l₁ + l₂ + l₃ + l₄ + l₅ + ... = ( Constant ) For string 1 l₁ + l₂ + l₃ = Constant For string 2 l₄ = Constant 35.6s