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Physics Lecture Review: Ideal Strings and Pulleys

Setup and Coordinate Choice

  • The scenario is analyzed on an inclined plane, using axes that are parallel and perpendicular to the plane (as opposed to using axes parallel and perpendicular to the ground).

  • The choice of axes is crucial as it simplifies the decomposition of forces, especially gravity, into components directly contributing to motion along the plane and forces perpendicular to it.

  • We consider the forces on a mass on the plane and how to resolve them into components along and perpendicular to the plane.

  • There may be one force pointing down (gravity) and possibly another force pulling the box (e.g., someone pulling it up the plane or dragging a box with a rope).

  • An example mentioned: dragging up a box, such as a U‑Haul task, to illustrate pulling a box up an incline.

Forces on a Mass on an Inclined Plane

  • Gravitational force acts downward: (\mathbf{F}_g = m\mathbf{g}). This force always points towards the center of the Earth.

  • Normal force acts perpendicular to the plane: (\mathbf{N}) with magnitude along the plane’s normal. This force arises from the contact between the mass and the surface, preventing the mass from penetrating the plane.

  • External pulling force can act along the plane (up the plane) with magnitude (F).

  • In many problems, there is no bungee cord; the mass is constrained to stay on the plane.

Decomposition of Forces into Parallel and Perpendicular Components

  • Gravity components:

    • Parallel to the plane: F_{g, \text{parallel}} = m g \sin\theta

    • Perpendicular to the plane: F_{g, \text{perp}} = m g \cos\theta

  • This decomposition is derived using basic trigonometry, where the angle \theta of the incline is also the angle between the gravitational force vector (mg) and the perpendicular axis. The component parallel to the plane is opposite to \theta from the (mg) vector when considering the right triangle formed, hence involving \sin\theta. The component perpendicular to the plane is adjacent to \theta, hence involving \cos\theta.

  • Normal force for a mass staying on the plane (no other vertical forces): N = m g \cos\theta

  • If an external force (F) is applied along the plane, it contributes to the parallel component:

    • Positive up the plane if (F) is directed up the plane. The direction is determined by the chosen positive axis along the incline.

Constraint and Idealizations: Why We Don’t Use Bungee Cords Here

  • The problem assumes the mass remains on the plane (a typical constraint in these analyses).

  • The mention of avoiding bungee cords emphasizes that the setup uses fixed, rigid constraints (the plane and possibly ropes) rather than elastic extensions that could alter the motion unexpectedly.

  • Using an elastic element like a bungee cord would introduce a variable spring force, making the problem significantly more complex due to its dependency on displacement and potentially leading to oscillatory motion. In these fundamental problems, we assume rigid interactions where the mass is constrained to the surface of the plane without deformation or elastic recoil.

Ideal String, Pulley, and Massless Points

  • Masses may be connected by an ideal string (inextensible and massless).

  • An ideal string is assumed to be inextensible (its length does not change, meaning connected masses move the same distance and thus have the same speed and magnitude of acceleration) and massless (its mass is negligible compared to the objects it connects, so no external forces are required to accelerate the string itself).

  • The string may go over an ideal pulley (a massless point) with negligible friction and inertia.

  • An ideal pulley is considered a massless point with negligible friction, which implies that it changes the direction of the string's tension without altering its magnitude or dissipating energy.

  • Idealization implications:

    • Tension is the same along the entire string (on both sides of the pulley).

    • There is no energy lost to turning or rotating the pulley.

    • The masses connected by the string accelerate in a coordinated way dictated by the string constraint.

  • The description suggests a configuration where one mass on the plane is connected via an ideal string to another mass (or masses) through an ideal pulley, so that forces are transmitted without loss of energy.

Forces Diagram and Considerations for the Mass on the Plane

  • The forces considered include:

    • Gravity (m g) downward, which contributes to both parallel and perpendicular components as above.

    • Normal force (N) perpendicular to the plane.

    • An external pulling force along the plane (the force tugging in a particular direction).

    • If present, friction along the plane (not explicitly mentioned in the transcript, but commonly discussed in these problems). If friction is present, it will always oppose the direction of relative motion or impending motion along the plane.

  • The line "So here, I have a force tugging this way" indicates an external force along the plane, with direction indicated by the chosen coordinate axis (positive up the plane).

Acceleration of the Mass on the Plane

  • Once the forces are resolved into components along the plane, the acceleration along the plane can be found from Newton’s second law along that axis:

    • For a single mass on a frictionless plane: m a = F - m g \sin\theta

    • If there is kinetic friction: m a = F - m g \sin\theta - f_k where (f_k = \mu_k N = \mu_k m g \cos\theta).

  • The sign convention is important: taking up the plane as positive, gravity contributes a negative component along the plane: ( - m g \sin\theta).

  • Therefore, the acceleration along the plane is

    a = \frac{F - m g \sin\theta - f_k}{m}

  • If there is no friction, this simplifies to

    a = \frac{F - m g \sin\theta}{m}.

Multi-Mass Systems Connected by an Ideal String

  • When multiple masses are connected by an ideal string over an ideal pulley, the rope enforces a constraint that couples their motions.

  • Key consequences of the ideal string and pulley:

    • Tension is the same throughout the string: typically denoted by (T).

    • The pulley is massless and frictionless in the idealization, so the string does not lose energy to rotation.

    • The accelerations of connected masses are related by the rope length constraint (e.g., if one mass moves a distance, the other moves correspondingly so the total rope length remains constant).

  • The rope length constraint ensures that the magnitude of acceleration for all connected masses is the same, assuming the string does not stretch and does not go slack. For instance, if mass m1 moves x distance up the incline, mass m2 will move x distance downwards (or upwards, depending on the setup).

  • A common two-mass setup (illustrative):

    • Mass on the plane (mass (m_1)) experiences along-plane forces: (T) (pulling up the plane), minus its gravity component along the plane, minus any friction.

    • A second mass (mass (m_2)) either hangs vertically or lies on another segment of the rope, with its equation depending on the orientation; for a hanging mass, the vertical motion yields: m_2 a = m_2 g - T.

    • Solve the coupled equations for (a) and (T) given (m_1, m_2, \theta, \mu_k) (if friction is present).

Real-World Relevance and Examples

  • Example alignment with everyday tasks: pulling a box up an incline (as with a U‑Haul scenario) uses exactly the same force decomposition and constraints (gravity components, normal force, and any pulling force along the plane).

  • The idealized rope and pulley model applies to many engineering systems (conveyors, cable cars, roller coasters when simplified) where energy losses are neglected for tractable analysis.

Practical Implications and Checks

  • If the external force becomes large enough that the normal force would become zero or negative, the constraint (the mass staying on the plane) would break and the problem would change fundamentally (the mass would leave the plane).

  • If the calculated normal force N becomes zero, it means the object is just about to lift off the surface. If N were to become negative in the calculations, it indicates that the object has already lost contact with the plane and the assumption of it staying on the plane is invalid; the problem would then involve projectile motion off the incline.

  • Idealizations (massless string, massless pulley, frictionless plane, etc.) simplify analysis but may not capture all real-world nuances (e.g., pulley inertia, rope stretch, friction variation).

  • The overall approach emphasizes:

    • Choosing coordinates along and perpendicular to the plane for clarity.

    • Decomposing forces into components relative to the plane.

    • Applying Newton’s laws along the constrained directions.

    • Using idealized constraints to relate accelerations in multi-mass systems.