Free Body Diagram Notes: Pulley-Block-Wedge Systems

Key Principles for Free-Body Diagrams (FBDs)

Identify every body in the system and draw its free-body diagram separately.
For each body, list all external forces on it:
- Weight: weight is always the force by Earth on the mass, written as $W = m g$ and acting downward.
- Normal forces: contact forces perpendicular to the contacting surface. If two bodies touch, there is a normal force on each body equal in magnitude and opposite in direction (Newton’s third law).
- Tension forces: along strings. If a string is light and inextensible and the pulley is ideal, the tension is the same in all portions of that string. If multiple strings are present, tensions in different strings can be different (denoted by different symbols such as $t, T, t1, t2,\ldots$ ).
- Pulley forces: when a pulley is present, forces from the string segments act on the pulley. If the pulley is attached to another body, these forces contribute to that body’s FBD unless you treat the pulley+body as a single system.
Direction conventions on inclined planes:
- Normal force is perpendicular to the plane.
- If a string lies along the plane, the tension is along the plane. The angle that the plane makes with the horizontal is typically denoted by $\theta$ .
- You may resolve components along the plane and perpendicular to the plane as needed, but remember the normal force is perpendicular to the plane.
Action–reaction pairs (Newton’s 3rd law):
- If surface A exerts a normal force on surface B, surface B exerts an equal and opposite normal force on surface A (e.g., ground on a block vs. block on the ground).
Pulley-attachment rule (the very important point):
- If a pulley is attached to a block, the forces acting on the pulley are transmitted to (and must be accounted for by) the block.
- If you place the pulley and block as a single system, you avoid dealing with the clamp forces that the attachment would otherwise imply.
- If you choose to separate the pulley from the block, you must include the clamp force (the reaction at the attachment) in the free-body diagram of the pulley or of the block, as appropriate.
Ideal pulley properties (massless, frictionless):
- The tensions on either side of an ideal pulley are equal in magnitude along the string.
- The pulley, being massless, has a net force of zero; any resulting vector sum of the tensions is balanced by the attachment (clamp) force.
- If a movable pulley has two rope segments pulling with tension $t1$ on opposite sides, the pulley experiences an upward force of $2 t1$ ; this is often balanced by the clamp/attachment.
Clamps and resultant forces on pulleys:
- For a pulley attached to a block, the clamp force must balance the vector sum of the tensions on the pulley.
- Example: if two perpendicular tension forces of magnitude $t$ act on a pulley, the clamp force has magnitude $F_{ ext{clamp}} = \sqrt{t^2 + t^2} = t\sqrt{2}.$ The direction is opposite to the resultant of the two tensions.
System choices:
- It is often simplest to treat a block + attached pulley as a single system to avoid tracking the internal clamp forces.
- If you do separate them, clearly label action/reaction pairs (e.g., normal forces N on each body, and their counterparts on the contact surface).
Step-by-step solving strategy (quick guide):
1) Draw the FBD for each body (block(s), pulleys, wedge, etc.).
2) Identify and label all forces: W, N, T (for each string), and any clamp forces if a pulley is separated.
3) Decide convenient axes (often along and perpendicular to an incline or along the string directions).
4) Resolve forces into components along chosen axes.
5) Apply Newton’s second law to each body: sum of forces along each axis equals mass times acceleration in that direction.
6) If multiple strings exist, remember a single string has the same tension along its length; different strings can carry different tensions.
7) If a pulley is massless and ideal, use the condition that the net force on the pulley is zero when considering the pulley in isolation, which helps relate tensions (e.g., for a movable pulley, end tension equals twice the segment tension).
Common mistakes to avoid:
- Forgetting to include the normal force from a surface that a body is in contact with.
- Confusing weight of one mass with a force on another body (weight acts on the mass due to Earth, not from another mass).
- Ignoring the clamp force when a pulley is not treated as part of the same system as the block to which it is attached.
- Mixing up action/reaction pairs or labeling two equal-and-opposite forces inconsistently.

Two-body system on an incline with a pulley (m1 and m2)

Setup (as described):
- Mass m1 sits on an inclined plane of angle $\theta$ . Free-body for m1 includes:
- Weight: $W{m1} = m1 g$ downward.
- Normal force from the plane: $N_1$ perpendicular to the plane.
- Tension along the string: $T$ along the plane (up the plane).
- Mass m2 is hanging, connected by the same string over a pulley. Free-body for m2 includes:
- Weight: $W{m2} = m2 g$ downward.
- Tension: $T$ upward along the string.
Tension direction note:
- The tension is along the string; the angle between the tension on m1 and the horizontal is the incline angle $\theta$ (i.e., the tension is parallel to the plane).
Forces on the pulley (in this simple setup):
- The string exerts a tension $T$ in two directions on the pulley: one along the plane (to the right) and one vertical (downwards).
- Because the string is a single, continuous, light, inextensible string, the tension magnitude is the same on both sides of the pulley: $|T|$ on each segment.
- The net force on the pulley is the vector sum of these two tensions; if the pulley is massless and attached to the block, the clamp must provide a force equal and opposite to this net force.
Resulting clamp force for the simple perpendicular tensions case:
- If the two tensions are perpendicular, the clamp force magnitude is $F_{ ext{clamp}} = \sqrt{T^2 + T^2} = T\sqrt{2}.$ The direction is opposite to the resultant of the two tensions.
Key takeaway:
- For a pulley attached to a block, treat the block+pulley as a single system to avoid the clamp force complexity; if separated, include the clamp force in the FBDs.

Important conceptual point: Pulley attached to a block

If a pulley is attached to a block, the forces on the pulley are transmitted to the block.
When you draw the FBD of the block alone, you must include the contact forces between the block and the pulley (the clamp/attachment forces) unless you have treated the block and pulley as a single system.
Example sequence discussed in class:
- A rectangular block of mass capital M sits on a frictionless horizontal floor. A light, frictionless pulley is attached to the block. A small mass m sits on top or adjacent, connected by a string that goes through the pulley.
- Free-body diagram of capital M (the block):
- Weight: $M g$ downward.
- Normal from ground: $N_{ ext{ground}}$ upward.
- Normal force by the small mass on the block: $N_1$ (the small mass pushes the block in some direction; the block pushes the small mass in the opposite direction).
- Tensions from the string acting on the pulley: two tensions, say, one to the right (along the string) and one downward (along the string) depending on the string path.
- Free-body diagram of the small mass: weight $m g$ downward; tension $T$ upward along the string; the small mass pushes the big block with a normal force equal and opposite to the normal exerted by the block on the small mass.
- Free-body diagram of the pulley (if drawn separately): the two tensions act on the pulley; the clamp must balance the vector sum of these tensions. If the pulley is massless and ideal, the net force on the pulley is zero, and the clamp force equals the vector sum of the tensions (e.g., magnitude $\sqrt{2} T$ for perpendicular directions).
Practical guidance:
- When possible, combine the block and attached pulley into a single system to simplify the force balance.
- If you must separate, label the action-reaction normal forces with distinct names (e.g., $N_1$ for the contact on the block by the ground, and the corresponding normal on the ground by the block) to avoid violating Newton’s third law.

Wedge/pulley system: Free-body diagrams with a wedge (capital M) and a small mass (m)

Setup description:
- A wedge (block) of mass capital M sits on a horizontal surface. The wedge has an incline at angle $\theta$ . A small mass m sits on the wedge or interacts with it via a string and a pulley attached to the wedge.
Free-body diagram for the wedge (capital M):
- Weight: $M g$ downward.
- Normal force from the ground: upward (let’s denote as $N_{ ext{ground}}$ ).
- Normal force from the small mass on the wedge: denoted as $N_1$ , perpendicular to the contact surface. This normal force on the wedge is due to the small mass pushing on the wedge.
- Forces from the pulley: the pulley is attached to the wedge and has tensions on it (for example, a tension $t$ along the string toward the right and a tension $t$ along the plane downward). Since the pulley is attached to the wedge, these tensions are effectively external forces on the wedge.
- The angle geometry: if a force is perpendicular to the incline, its direction is determined by the normal to the wedge surface; the tension directions depend on the string path relative to the incline.
Free-body diagram for the small mass (m):
- Weight: $m g$ downward.
- Tension: along the string, typically upward along the plane, magnitude equal to the tension in the string (denoted by $t$ or another symbol depending on the string).
- Normal reaction from the wedge: perpendicular to the contact surface. This normal force is the counterpart to the wedge’s normal force on the small mass (action–reaction pair).
- Action/reaction consistency: the small mass pushes on the wedge with a normal force equal and opposite to the normal force from the wedge on the small mass.
Free-body diagram for the pulley (attached to the wedge):
- If the string goes around the pulley, there are tensions along the two string segments acting on the pulley (e.g., one to the right, one down the incline).
- The net force on the pulley is the vector sum of these tensions; the clamp (attachment) to the wedge must provide the equal and opposite reaction to keep the pulley attached.
Key note from the lecture for this setup:
- Normal forces come in action–reaction pairs across surfaces (e.g., between mass and wedge, wedge and ground).
- The pulley exerts forces on the wedge through the attachment; you can either include them explicitly if you separate the pulley, or treat the wedge+pulley as a single system to avoid explicit clamp calculations.

Three masses with a pulley system: m1, m2, m3 and pulleys

General approach:
- Draw FBDs for m1, m2, m3, and for each pulley separately if you are not treating the whole assembly as a single system.
- Identify all forces on each mass:
- For the block m1: weight $m_1 g$ downward; normal force from the ground or from the supporting surface; normal force from the adjoining block (e.g., from m2) along the contact; a tension from the string along the connection path.
- For the block m2: weight $m_2 g$ downward; normal forces from adjacent surfaces; tension from the string(s) attached to m2; and any normal interactions with adjacent blocks.
- For the block m3: weight $m_3 g$ downward; tension from its string as appropriate; any normals from adjacent surfaces.
- For the pulley(s): if drawn, there are tensions from the strings on each side of the pulley. If the pulley is movable, the net force on the pulley must be balanced by the clamp force if you treat the pulley as a separate body.
Specific points highlighted in the transcript:
- The wedge-like or stacked-block setup can involve action–reaction normals at multiple contact surfaces (
- Normal from ground on the big block,
- Normal from small blocks on the big block,
- Normal from block on the plank, etc.).
- If there are multiple strings with different tensions, each string carries a potentially different tension (e.g., a string with tension $t1$ and another with tension $t2$ ).
- In some configurations with a single string looping around several pulleys, the tensions along the string are the same everywhere on that string, but different strings can carry different tensions.
Example consequence discussed:
- In a certain configuration with three pulleys and one string connecting the smaller mass and the larger block, there may be four tension forces pulling on the large block (effectively four segments of the same string contributing forces on the block). If the tension in that string is $T$ , then the large block experiences four forces of magnitude $T$ in different directions.
- The free-body diagram of the pulley(s) helps determine how many tension forces act on the blocks and how the tensions relate to each other (e.g., movable pulleys cause relations like $T = 2 t_1$ for the end-to-end tension versus the segment tension in an ideal pulley system).
Important relation to remember in multi-pulley systems:
- For an ideal, massless, frictionless movable pulley with a rope of tension $t1$ on each side, the load ( pulley ) experiences an upward force of $2 t1$ , and the support (clamp) supplies the balancing force to keep the pulley in equilibrium if needed.

Special notes on tension and system simplifications

Tension along a single string is uniform along that string in an ideal rope: the same magnitude on every segment of that string.
If multiple strings are present, each string can have its own tension (denoted by different symbols, e.g., $t1, t2, T,\ldots$ ).
In a pulley-block system, if a pulley belongs to a block and is essential to the mechanism, it is often easiest to treat the block and pulley as a single rigid body. This avoids dealing with the clamp force at the attachment; otherwise you must include the clamp force in the pulley’s FBD.
When a pulley is massless and frictionless:
- The net force on the pulley is zero; the tensions on the two sides sum to zero when the directions are opposite, and any nonzero resultant is balanced by the clamp force in the attachment.
If a system includes several surfaces in contact, always identify the action–reaction pairs for each contact (e.g., the normal force on each body from the other, and the counterpart on the other body).

Quick recap of key formulas to remember

Weight and normal force:
- $W = m g$
- Normal forces are perpendicular to the contacting surfaces.
Tension along a string (ideal rope):
- Magnitude is the same along the entire string; if there are multiple strings, tensions can differ: $T1, T2, \ldots$
Forces on a pulley attached to a block (example with two perpendicular tensions):
- If the tensions are perpendicular and equal in magnitude $T$ , clamp force magnitude is $F_{ ext{clamp}} = \sqrt{T^2 + T^2} = T\sqrt{2}.$
Movable pulley relation (ideal, massless, frictionless):
- End-to-end tension on the rope related to the segment tension by $T = 2 t1$ (or equivalently, the load experiences a force $2 t1$ from the two rope segments).
System approach guidelines:
- Prefer treating blocks and attached pulleys as a single system when possible.
- If separate, clearly label and account for clamps/attachments.

Practice tips for exam preparation

Practice drawing FBDs for the same physical setup from different perspectives (block-only, pulley-only, and block+pulley as a composite) to ensure you account for all forces.
Always label forces clearly (e.g., $N{ ext{ground}}$ , $N{1}$ between block and mass, $T$ for tension along a string, etc.).
When dealing with wedges and inclines, consistently denote the angle $\theta$ and consider components along and perpendicular to the plane if needed.
Check that action–reaction pairs appear on the appropriate bodies and that the sum of forces on a massless pulley is balanced by the attachment force.
After drawing FBDs, proceed to resolve forces along chosen axes and apply Newton’s second law to each body, then solve the resulting system of equations consistently.