## Step-by-Step Solution Let's break down the problem: ### **Given Data** - Two hydraulic cylinders at points C, each applying force \( F \). - The total mass being lifted (platform + workers) = \( m \). - The scissor mechanism's mass is negligible. - **Goal:** Find the platform's **upward acceleration \( a \)**, and show it **does not depend** on dimension \( b \) or angle \( \theta \). --- ### **Step 1: Free Body Diagram** - The platform is lifted vertically. - Two cylinders push up at points C. - The total upward force = \( 2F \). - The total downward force (weight) = \( mg \). --- ### **Step 2: Newton's Second Law** The **net force** on the platform: \[ \text{Net Force} = \text{Upward Force} - \text{Downward Force} = 2F - mg \] By Newton's Second Law: \[ \text{Net Force} = m a \] So, \[ m a = 2F - mg \] \[ a = \frac{2F - mg}{m} \] --- ### **Step 3: Check for Dependence on \( b \) and \( \theta \)** - **Force \( F \)** is the force *produced by each cylinder*, not a component of some force. - The **vertical component of the force** from both cylinders is directly lifting the platform. - The mechanical arrangement (scissors, angles, linkage length \( b \)) does not affect the relationship between the vertical force applied and the acceleration, because the linkage is massless and frictionless (as per the problem statement). **Therefore, the acceleration \( a \) is independent of both \( b \) and \( \theta \).** --- ### **Step 4: Final Answer** #### **Upward acceleration of the platform:** \[ \boxed{a = \frac{2F}{m} - g} \] #### **Independence from \( b \) and \( \theta \):** - \( a \) **only depends** on the applied force \( F \) and mass \( m \). - **No terms involving \( b \) or \( \theta \)** appear in the final formula. --- ## **Summary Table** | Variable | Meaning | Appears in \( a \)? | |----------|--------------------------------|---------------------| | \( F \) | Force from each cylinder | **Yes** | | \( m \) | Mass of platform + workers | **Yes** | | \( g \) | Gravity | **Yes** | | \( b \) | Linkage dimension | No | | \( \theta \) | Linkage angle | No | --- ## **Conclusion** - The acceleration of the platform is: \[ \boxed{a = \frac{2F}{m} - g} \] - This acceleration is **independent of both the linkage dimension \( b \) and the angle \( \theta \)**. If you have follow-up questions about the mechanics or the reasoning, please ask!

# Analysis of Portable Work Platform Acceleration ## Given Data - Two hydraulic cylinders apply a force \( F \) each. - Total mass of platform and workers: \( m \). - Objective: Determine upward acceleration \( a \) and show its independence from dimension \( b \) and angle \( \theta \). --- ## Step 1: Free Body Diagram - **Upward Force:** Total force from cylinders: \( 2F \). - **Downward Force:** Weight of the platform: \( mg \). --- ## Step 2: Apply Newton's Second Law Using Newton's Second Law: \[ \text{Net Force} = \text{Upward Force} - \text{Downward Force} \] \[ 2F - mg = ma \] Rearranging gives: \[ a = \frac{2F - mg}{m} \] --- ## Step 3: Analyzing Dependence on \( b \) and \( \theta \) - The force \( F \) is produced directly by the hydraulic cylinders and is independent of the linkage dimensions or angle. - The mechanical design (scissor lift) translates the force upward without affecting the total force balance. ### Conclusion on Independence The acceleration \( a \) does not change with variations in: - **Dimension \( b \)** - **Angle \( \theta \)** --- ## Step 4: Final Expression for Acceleration The upward acceleration of the platform can be summarized as: \[ \boxed{a = \frac{2F}{m} - g} \] --- ## Summary Table | Variable | Description | Appears in \( a \)? | |----------|-----------------------------------|---------------------| | \( F \) | Force from each hydraulic cylinder | **Yes** | | \( m \) | Total mass of the platform | **Yes** | | \( g \) | Acceleration due to gravity | **Yes** | | \( b \) | Dimension of the linkage | No | | \( \theta \) | Angle of the linkage | No | --- ## Conclusion - The acceleration of the platform is defined by: \[ \boxed{a = \frac{2F}{m} - g} \] - This result confirms that the upward acceleration is independent of both the linkage dimension \( b \) and the angle \( \theta \). For any further clarifications or questions, feel free to ask!