Physics Study Notes: Static Equilibrium, Friction, and Mechanical Systems

Static Equilibrium Analysis of a Vehicle on an Incline (Figure E)

Scenario Overview: A car is positioned on a significantly slippery hillside. To prevent the vehicle from sliding down due to gravity, the owner has secured it using a cable.
Numerical Parameters:
- Mass of the car ( $m$ ): $1400\,kg$
Task Components:
- (a) Force Mapping and Coordinate Systems: Sketch every force acting upon the car. This includes Gravity ( $F_g$ ), the Normal force ( $N$ ) perpendicular to the surface of the hill, and the Tension force ( $T$ ) exerted by the cable. Coordinate axes must be defined, typically with the $x$ -axis parallel to the incline and the $y$ -axis perpendicular to it.
- (b) Force Components: Determine the vector components for all identified forces relative to the chosen coordinate system. Gravity is resolved into:
  - $F_{gx} = m \times g \times \sin(\theta)$
  - $F_{gy} = m \times g \times \cos(\theta)$
- (c) Equilibrium Equations: Formulate the mathematical conditions for static equilibrium along both coordinate directions:
  - $\sum F_x = 0$
  - $\sum F_y = 0$
- (d) Tension Calculation: If the interface between the road and the tires is frictionless ( $\mu = 0$ ), calculate the tension ( $T$ ) required to maintain the car's position. This calculation relies on the balance between tension and the down-slope component of gravity:
  - $T = m \times g \times \sin(\theta)$

Static Equilibrium of Connected Blocks on a Horizontal Surface (Figure F)

System Configuration: Two blocks, $m_1$ and $m_2$ , are connected via a massless string. The string passes over a frictionless pulley.
Assigned Values:
- Mass of block 1 ( $m_1$ ): $45\,kg$
- Mass of block 2 ( $m_2$ ): $12\,kg$
- Coefficient of static friction ( $\mu_s$ ) between $m_1$ and the table: $0.45$
Analysis Requirements:
- (a) Stability Assessment: Determine if the system remains in static equilibrium. This requires comparing the weight of the hanging mass ( $m_2 \times g$ ) against the maximum possible static frictional force ( $f_{s,max} = \mu_s \times m_1 \times g$ ) acting on the table-bound mass.
- (b) Tension Determination: Calculate the magnitude of the tension in the string during the system's current state.

Experimental Determination of Kinetic Friction for a Sled

Initial Dynamic State:
- Applied Horizontal Force ( $F$ ): $20\,N$
- Resultant Acceleration ( $a$ ): $0.40\,m/s^2$
- Surface: Level trail
Revised State Conditions:
- An additional mass ( $m_{extra}$ ) of $4.5\,kg$ is added to the sled.
- The same horizontal force ( $20\,N$ ) is applied.
- Observation: The force is now "just barely" enough to keep the sled in motion, implying a state of constant velocity where acceleration is effectively zero ( $a = 0$ ).
Primary Goal: Calculate the coefficient of kinetic friction ( $\mu_k$ ) between the sled and the trail using the established change in mass and the resulting change in the system's acceleration.

Maximum Mass Thresholds for Static Equilibrium on an Incline (Figure G)

System Parameters:
- Block 1 mass ( $m_1$ ): $5.0\,kg$
- Incline angle ( $\theta$ ): $35^\circ$
- Coefficient of static friction ( $\mu_s$ ) between block 1 and the incline: $0.25$
Objective: Determine the largest possible value for the hanging mass ( $m_2$ ) such that the system remains at rest (static equilibrium). This occurs at the threshold where block 1 is on the verge of sliding up the incline, meaning the static friction force is directed down the ramp at its maximum value:
- $m_2 \times g = m_1 \times g \times \sin(\theta) + \mu_s \times m_1 \times g \times \cos(\theta)$

Chapter 5 Curricular Outline

5.1: Force, Displacement, and Work
5.2: Kinetic Energy and the Work-Energy Theorem
5.3: Potential Energy and Conservation of Energy
5.4: More Potential Energy Functions
5.5: Conservative and Nonconservative Forces; Conservation of Mechanical Energy
5.6: The Nature of Forces: What is Work?
5.7: Power

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Static Equilibrium Analysis of a Vehicle on an Incline (Figure E)

Scenario Overview: A car is positioned on a significantly slippery hillside. To prevent the vehicle from sliding down due to gravity, the owner has secured it using a cable.
Numerical Parameters:
- Mass of the car ( $m$ ): $1400\,kg$
Task Components:
- (a) Force Mapping and Coordinate Systems: Sketch every force acting upon the car. This includes Gravity ( $F_g$ ), the Normal force ( $N$ ) perpendicular to the surface of the hill, and the Tension force ( $T$ ) exerted by the cable. Coordinate axes must be defined, typically with the $x$ -axis parallel to the incline and the $y$ -axis perpendicular to it.
- (b) Force Components: Determine the vector components for all identified forces relative to the chosen coordinate system. Gravity is resolved into:
  - $F_{gx} = m \times g \times \sin(\theta)$
  - $F_{gy} = m \times g \times \cos(\theta)$
- (c) Equilibrium Equations: Formulate the mathematical conditions for static equilibrium along both coordinate directions:
  - $\sum F_x = 0$
  - $\sum F_y = 0$
- (d) Tension Calculation: If the interface between the road and the tires is frictionless ( $\mu = 0$ ), calculate the tension ( $T$ ) required to maintain the car's position. This calculation relies on the balance between tension and the down-slope component of gravity:
  - $T = m \times g \times \sin(\theta)$

Static Equilibrium of Connected Blocks on a Horizontal Surface (Figure F)

System Configuration: Two blocks, $m1$ and $m2$ , are connected via a massless string. The string passes over a frictionless pulley.
Assigned Values:
- Mass of block 1 ( $m_1$ ): $45\,kg$
- Mass of block 2 ( $m_2$ ): $12\,kg$
- Coefficient of static friction ( $\mus$ ) between $m1$ and the table: $0.45$
Analysis Requirements:
- (a) Stability Assessment: Determine if the system remains in static equilibrium. This requires comparing the weight of the hanging mass ( $m2 \times g$ ) against the maximum possible static frictional force ( $f{s,max} = \mus \times m1 \times g$ ) acting on the table-bound mass.
- (b) Tension Determination: Calculate the magnitude of the tension in the string during the system's current state.

Experimental Determination of Kinetic Friction for a Sled

Initial Dynamic State:
- Applied Horizontal Force ( $F$ ): $20\,N$
- Resultant Acceleration ( $a$ ): $0.40\,m/s^2$
- Surface: Level trail
Revised State Conditions:
- An additional mass ( $m_{extra}$ ) of $4.5\,kg$ is added to the sled.
- The same horizontal force ( $20\,N$ ) is applied.
- Observation: The force is now "just barely" enough to keep the sled in motion, implying a state of constant velocity where acceleration is effectively zero ( $a = 0$ ).
Primary Goal: Calculate the coefficient of kinetic friction ( $\mu_k$ ) between the sled and the trail using the established change in mass and the resulting change in the system's acceleration.

Maximum Mass Thresholds for Static Equilibrium on an Incline (Figure G)

System Parameters:
- Block 1 mass ( $m_1$ ): $5.0\,kg$
- Incline angle ( $\theta$ ): $35^\circ$
- Coefficient of static friction ( $\mu_s$ ) between block 1 and the incline: $0.25$
Objective: Determine the largest possible value for the hanging mass ( $m_2$ ) such that the system remains at rest (static equilibrium). This occurs at the threshold where block 1 is on the verge of sliding up the incline, meaning the static friction force is directed down the ramp at its maximum value:
- $m2 \times g = m1 \times g \times \sin(\theta) + \mus \times m1 \times g \times \cos(\theta)$