Coordinate Setup for Multi-Mass Systems: Friction, Tension, and Acceleration
Coordinate Setup and Notation
Define a system with multiple masses (e.g., three blocks) to set up coordinates clearly.
For block 1: mass is m1 and its position is denoted by coordinates with subscripts:
x1, y1, z1 for block 1
For block 2: mass is m2 and position (x2, y2, z2)
For block 3: mass is m3 and position (x3, y3, z3)
The coordinates should allow you to say exactly which position you’re talking about (e.g., block 1’s position is (x1, y1, z1))
The choice of coordinate axes is arbitrary but must be used consistently; motion direction relative to your chosen axes determines signs of components
Example: if the mass falls downward, you can choose positive y to point downward, but the convention must be understood and kept consistent
If you choose positive y downward, and the system accelerates downward, then the acceleration component a1y (and/or others) will come out positive accordingly
Signs of acceleration reflect the chosen coordinate directions:
If the motion is in the positive direction, the corresponding acceleration component is positive
If the motion is in the negative direction, the corresponding acceleration component is negative
You must be precise about component notation: for block i, use a{i x}, a{i y}, a_{i z} to specify each block’s acceleration components along each axis
The net force components correspondingly have subscripts: F{i x}, F{i y}, F_{i z}
The net force vector on each block drives its acceleration via Newton’s second law in component form
Component-wise Newton’s Second Law for Each Block
For each block i, apply
egin{aligned}
ext{Sum of forces along } x:
&\sum F{i x} = mi \, a{i x}, \ ext{Sum of forces along } y: &\sum F{i y} = mi \, a{i y}, \
ext{Sum of forces along } z:
&\sum F{i z} = mi \, a_{i z}.
\end{aligned}
The forces contributing to these sums can include:
Tension forces (e.g., T1, T2) with their proper directions in the chosen coordinate system
Friction forces (static or kinetic), which depend on the normal forces and the coefficient of friction, and act to oppose motion up to a maximum static value
Gravity, normal forces, and any other external forces present in the setup
Important point: you must specify the direction of each force component; if a force acts in the negative direction relative to the chosen axis, its component is negative in the equation
In this kind of system, you will typically obtain a set of coupled equations involving the accelerations {a{1x}, a{1y}, a{2x}, a{2y}, a{3x}, a{3y}, …} and the tension variables (e.g., T1, T2)
The goal is to form enough independent equations to solve for the unknowns (accelerations and tensions). Often, you’ll eliminate tensions by combining equations (e.g., subtracting some, adding others)
A common strategic step is to write down the force balance equations for each block in each relevant direction, then manipulate them to isolate the unknown acceleration a (the system’s overall acceleration along the direction of motion)
Practical approach emphasis: focus on what you know and what you don’t know, then decide whether you can obtain a directly or whether you must first express tensions in terms of a or other knowns
Elimination Strategy: Solving for the Acceleration a
A typical approach is to subtract or combine block equations to eliminate the internal tension forces T1 and T2
By doing so, you obtain an equation that directly relates the system acceleration a to the external forces and the masses
Example outline (generic):
Write the component equations for each block along the axis of motion(s) where the accelerations are defined
Subtract (or add) the equations to cancel out T1 and T2
Solve the resulting equation for the acceleration a
Once a is known, substitute back to determine the tensions T1, T2 (and any frictional forces) if needed
After obtaining a, you can then plug in the actual masses and other known quantities to compute the numerical value of the acceleration
A useful overarching result: when internal forces (like tensions) cancel in the sum of the equations for all blocks along the direction of motion, the total external force equals the total mass times the system acceleration, i.e.,
ext{(Sum of external forces along a given direction)} \ = (m1 + m2 + m_3) \, aThis is a manifestation of the conservation of momentum for the system when only external forces act on the center-of-mass motion along that direction
If you have several directions, you analyze the direction of motion and apply the same elimination idea along that axis
Case Analysis: Static and Kinetic Friction Scenarios (Five Cases Mentioned)
The following cases are described in the transcript to understand when the system does not move vs. when movement occurs:
Case 1: If tension T2 is bigger in its direction, there might be enough static friction in that direction to stop the system from starting to move
Case 2: The second possibility is that tension is larger in the other direction, but static friction in that direction is enough to prevent movement
Case 3: The tensions are equal. If this common tension is less than the sum of the opposing friction thresholds (i.e., less than friction plus the other tension), the system stays stationary
The above cases describe situations in which static friction is sufficient to prevent motion
Practical interpretation:
Static friction can resist a range of net tangential forces up to a maximum value; if the net driving tendency is within this range, the blocks do not slip
If the net driving tendency exceeds the maximum static friction, the system will begin to move and kinetic friction will come into play
Additional note from the transcript about resolving directions and directions of motion:
Consider a scenario with blocks moving in different directions (e.g., one block moving in negative y, another in positive x, etc.). In such a setup, the T2 tension terms may cancel when you add/subtract the appropriate equations, which helps isolate the acceleration a
Case 4/5 idea (from the transcript’s discussion about cancellation):
If you choose accelerations in three directions such that the T2 terms cancel when combining the block equations, you can rearrange to solve for a using the remaining external forces and masses
The transcript suggests:
Block is going in the negative y direction. This block is going in the positive x direction. This block is going in the positive y direction. The T2 terms will cancel.This leads to using a combination of the three equations to eliminate T2 and then T1 as well, leaving an expression for a
Takeaway: Case analysis helps determine whether the system remains at rest or moves, and guides how you should combine equations to solve for the acceleration
Practical workflow and mental model
Step 1: Set up coordinates clearly and assign x, y (and z if needed) directions for all blocks; label accelerations a{i x}, a{i y}, a_{i z}
Step 2: Write down the component equations for each block along each axis that matters (x, y, z) with the correct sign conventions
Step 3: Include external forces (gravity, normal forces, friction) and internal forces (tensions T1, T2) with proper directions
Step 4: Decide whether you will solve by elimination of tensions (subtracting or adding equations) or by solving the full system simultaneously
Step 5: If static friction is a factor, check whether the net driving forces exceed the static friction threshold; if not, set accelerations to zero and stop further calculation for motion
Step 6: If motion occurs, use the elimination method to obtain a expression for the acceleration a, then back-substitute to find tensions and friction forces if needed
Step 7: Plug in actual masses and any given numerical values to obtain a concrete numerical answer
Summary and Takeaways
Coordinate setup is crucial for clarity when multiple masses are involved
Always keep track of sign conventions and be explicit about which component you are solving for
Use a vector approach and component equations to apply Newton’s second law to each block
Tensions can often be eliminated by combining the block equations; this is a powerful method to isolate the system’s overall acceleration
Static friction analysis determines whether the system remains at rest or moves; if the driving forces lie within the static friction limits, the system does not move
The “sum of external forces” approach provides a quick check: if internal forces cancel in the direction of motion, the acceleration depends only on external forces and the total mass
Practice by working through the elimination of T1 and T2 and verify that results are consistent with the chosen coordinate directions and friction constraints
Coordinate Setup and Notation
Define a system with multiple masses (e.g., three blocks) to set up coordinates clearly.
For block 1: mass is m1 and its position is denoted by coordinates with subscripts:
x1, y1, z1 for block 1
For block 2: mass is m2 and position (x2, y2, z2)
For block 3: mass is m3 and position (x3, y3, z3)
The coordinates should allow you to say exactly which position you’re talking about (e.g., block 1’s position is (x1, y1, z1))
The choice of coordinate axes is arbitrary but must be used consistently; motion direction relative to your chosen axes determines signs of components.
It is crucial to establish a single, consistent coordinate system for all interconnected blocks, especially when their motions are related (e.g., by a rope or pulley).
Example: if the mass falls downward, you can choose positive y to point downward, but the convention must be understood and kept consistent
If you choose positive y downward, and the system accelerates downward, then the acceleration component a1y (and/or others) will come out positive accordingly
Signs of acceleration reflect the chosen coordinate directions:
If the motion is in the positive direction, the corresponding acceleration component is positive
If the motion is in the negative direction, the corresponding acceleration component is negative
You must be precise about component notation: for block i, use a{i x}, a{i y}, a_{i z} to specify each block’s acceleration components along each axis
The net force components correspondingly have subscripts: F{i x}, F{i y}, F_{i z}
The net force vector on each block drives its acceleration via Newton’s second law in component form
Component-wise Newton’s Second Law for Each Block
For each block i, apply
\begin{aligned}
\text{Sum of forces along } x: &\sum F{i x} = mi \, a{i x}, \ \text{Sum of forces along } y: &\sum F{i y} = mi \, a{i y}, \
\text{Sum of forces along } z: &\sum F{i z} = mi \, a_{i z}.
\end{aligned}Before applying Newton's Second Law, it is essential to draw a free-body diagram for each block, clearly showing all forces acting on it. This helps in identifying all forces and their directions.
The forces contributing to these sums can include:
Tension forces (e.g., T1, T2) with their proper directions in the chosen coordinate system
Friction forces (static or kinetic), which depend on the normal forces and the coefficient of friction, and act to oppose motion up to a maximum static value.
The type of friction (static or kinetic) depends on whether the system is at rest or in motion.
Gravity, normal forces, and any other external forces present in the setup
Important point: you must specify the direction of each force component; if a force acts in the negative direction relative to the chosen axis, its component is negative in the equation
In this kind of system, you will typically obtain a set of coupled equations involving the accelerations a{1x}, a{1y}, a{2x}, a{2y}, a{3x}, a{3y} and the tension variables (e.g., T1, T2)
The goal is to form enough independent equations to solve for the unknowns (accelerations and tensions). Often, you’ll eliminate tensions by combining equations (e.g., subtracting some, adding others)
A common strategic step is to write down the force balance equations for each block in each relevant direction, then manipulate them to isolate the unknown acceleration a (the system’s overall acceleration along the direction of motion)
Practical approach emphasis: focus on what you know and what you don’t know, then decide whether you can obtain a directly or whether you must first express tensions in terms of a or other knowns
Elimination Strategy: Solving for the Acceleration a
A typical approach is to subtract or combine block equations to eliminate the internal tension forces T1 and T2.
This cancellation occurs because internal forces like tension—acting between connected blocks—are action-reaction pairs according to Newton's Third Law. When equations for interconnected blocks are summed along the direction of motion, these internal forces often appear with opposite signs and thus cancel out, allowing the system's acceleration to be determined solely from the external forces.
By doing so, you obtain an equation that directly relates the system acceleration a to the external forces and the masses
Example outline (generic):
Write the component equations for each block along the axis of motion(s) where the accelerations are defined
Subtract (or add) the equations to cancel out T1 and T2
Solve the resulting equation for the acceleration a
Once a is known, substitute back to determine the tensions T1, T2 (and any frictional forces) if needed
After obtaining a, you can then plug in the actual masses and other known quantities to compute the numerical value of the acceleration
A useful overarching result: when internal forces (like tensions) cancel in the sum of the equations for all blocks along the direction of motion, the total external force equals the total mass times the system acceleration, i.e.,
\text{(Sum of external forces along a given direction)} \ = (m1 + m2 + m_3) \, a
This is a manifestation of the conservation of momentum for the system when only external forces act on the center-of-mass motion along that direction
If you have several directions, you analyze the direction of motion and apply the same elimination idea along that axis
Case Analysis: Static and Kinetic Friction Scenarios (Five Cases Mentioned)
The following cases are described in the transcript to understand when the system does not move vs. when movement occurs:
Case 1: If tension T2 is bigger in its direction, there might be enough static friction in that direction to stop the system from starting to move
Case 2: The second possibility is that tension is larger in the other direction, but static friction in that direction is enough to prevent movement
Case 3: The tensions are equal. If this common tension is less than the sum of the opposing friction thresholds (i.e., less than friction plus the other tension), the system stays stationary
The above cases describe situations in which static friction is sufficient to prevent motion
Practical interpretation:
Static friction can resist a range of net tangential forces up to a maximum value; if the net driving tendency is within this range, the blocks do not slip.
The maximum static friction force is given by \mus N, where \mus is the coefficient of static friction and N is the normal force. If the net driving force along the surface is less than or equal to \mu_s N, the object remains static, and the static friction force adjusts to oppose the applied force.
If the net driving tendency exceeds the maximum static friction, the system will begin to move and kinetic friction will come into play.
If the net driving force exceeds \mus N, the object begins to move, and the friction force transitions to kinetic friction, which is typically constant and given by \muk N, where \muk is the coefficient of kinetic friction (\muk < \mu_s).
Additional note from the transcript about resolving directions and directions of motion:
Consider a scenario with blocks moving in different directions (e.g., one block moving in negative y, another in positive x, etc.). In such a setup, the T2 tension terms may cancel when you add/subtract the appropriate equations, which helps isolate the acceleration a
Case 4/5 idea (from the transcript’s discussion about cancellation):
If you choose accelerations in three directions such that the T2 terms cancel when combining the block equations, you can rearrange to solve for a using the remaining external forces and masses
The transcript suggests:
Block is going in the negative y direction. This block is going in the positive x direction. This block is going in the positive y direction. The T2 terms will cancel.This leads to using a combination of the three equations to eliminate T2 and then T1 as well, leaving an expression for a
Takeaway: Case analysis helps determine whether the system remains at rest or moves, and guides how you should combine equations to solve for the acceleration
Practical workflow and mental model
Step 1: Draw Free-Body Diagrams (FBDs) for each block, clearly indicating all forces (gravity, normal, tension, friction, external applied forces) and their directions.
Step 2: Establish a Consistent Coordinate System for all blocks. Define positive x, y (and z if needed) directions for each block and ensure consistency for interconnected elements.
Step 3: Identify Knowns and Unknowns. List all given masses, coefficients of friction, angles, and forces. Identify the variables you need to solve for (e.g., accelerations, tensions, normal forces).
Step 4: Write down the component equations (\sum F = ma) for each block along each axis that matters (x, y, z) with the correct sign conventions.
Step 5: Determine the State of Motion:
If static friction is a factor, calculate the net driving force along the potential direction of motion assuming no motion (i.e., treating friction as static and adjustable).
Compare this net driving force to the maximum static friction (\mu_s N).
If the net driving force is less than or equal to the maximum static friction, the system remains at rest (accelerations are zero). The static friction force will be equal to the net driving force.
If the net driving force exceeds the maximum static friction, the system will accelerate. Proceed to the next step, using kinetic friction (\mu_k N) in your equations.
Step 6: If motion occurs, use the elimination method (subtracting or adding equations) to obtain an expression for the acceleration a.
Step 7: Back-substitute the value of a (or its expression) into the original force equations to find tensions and friction forces if needed.
Step 8: Plug in actual masses and any given numerical values to obtain a concrete numerical answer for accelerations, tensions, etc.
Summary and Takeaways
Coordinate setup and Free-Body Diagrams are crucial for clarity when multiple masses are involved
Always keep track of sign conventions and be explicit about which component you are solving for
Use a vector approach and component equations to apply Newton’s second law to each block
Tensions can often be eliminated by combining the block equations; this is a powerful method to isolate the system’s overall acceleration by leveraging Newton's Third Law for internal forces
Static friction analysis determines whether the system remains at rest or moves; if the driving forces lie within the static friction limits, the system does not move. If they exceed it, kinetic friction applies.
The “sum of external forces” approach provides a quick check: if internal forces cancel in the direction of motion, the acceleration depends only on external forces and the total mass
Practice by working through the elimination of T1 and T2 and verify that results are consistent with the chosen coordinate directions and friction constraints