Simple Machines

Cheatsheet Content

### Simple Machines: Introduction Devices used to do work easily and quickly with less effort (or force) are called simple machines. They change the magnitude, direction, or point of application of force. **How they help:** 1. **Multiply force:** Applying a small force at one point can produce a large force at another. 2. **Change direction:** Alter the direction of the applied force. 3. **Change point of application:** Move the point where force is applied. 4. **Increase speed:** Achieve higher speeds in doing work. **Examples:** 1. The Lever 2. The Inclined Plane 3. The Wedge 4. The Pulley 5. The Screw 6. The Wheel and the Axle **Two sides of a simple machine:** 1. **Input side:** Where effort or energy is supplied. 2. **Output side:** Where useful work is done by the machine. ### Important Terms - **Effort (E):** The external force applied at a convenient point of a machine to overcome the force (load). Measured in Newtons (N). - **Load (L):** The force to be overcome by the machine by applying effort. Measured in Newtons (N). - **Mechanical Advantage (M.A):** The ratio of the load to the effort. $$ M.A = \frac{\text{Load}}{\text{Effort}} $$ - It has no unit. - **Velocity Ratio (V.R):** The ratio of the distance moved by the effort to the distance moved by the load in the same time. $$ V.R = \frac{\text{Distance moved by the effort}}{\text{Distance moved by the load}} $$ - It has no unit. - **Efficiency ($\eta$):** The ratio of the useful work done by the machine (output work) to the work done on the machine (called input work). $$ \eta = \frac{\text{Output work}}{\text{Input work}} $$ - Expressed as a percentage: $$ \text{Percentage efficiency} = \eta \times 100\% = \frac{\text{Output work}}{\text{Input work}} \times 100\% $$ - For an ideal machine, efficiency is 100% or 1. - For actual machines, efficiency is always less than 100% due to friction. **Relation between Mechanical Advantage, Velocity Ratio and Efficiency:** For an ideal machine, M.A = V.R. $$ \eta = \frac{\text{M.A}}{\text{V.R}} $$ ### Levers A rigid rod free to turn about a fixed point called the **fulcrum (F)**. **Types of Levers:** Levers are classified into three classes based on the relative positions of the effort (E), load (L), and fulcrum (F). #### Class I Levers - **Fulcrum (F)** is in between the **Effort (E)** and the **Load (L)**. - Examples: Seawaw, a pair of scissors, a crowbar, dew hammer, a beam of common balance, a pair of pliers etc. - **M.A > 1:** If F is closer to L than E (effort arm > load arm). - **M.A ### Inclined Plane A machine used to raise heavy loads with less effort or force. - It is found that less force is needed to raise the load along the inclined plane than to raise it directly upward to the same height. - However, work done on the load is the same in both cases. ### Pulleys A pulley is a wooden or metallic wheel with a grooved rim on an axle or shaft passing through its centre. The pulley rotates freely about the axle and the axle is fixed in a frame or a block. The effort is applied to a rope passing over the grooved rim of the pulley. #### Types of Pulleys 1. Single Fixed Pulley 2. Single Movable Pulley 3. Combination of Fixed and Movable Pulley 4. Combination of Pulleys: Block and Tackle System #### Single Fixed Pulley A single fixed pulley is a pulley whose block is fixed to a rigid support. It acts as a Class I lever. In this pulley, the effort arm equals the load arm. **Mechanical advantage of a single fixed pulley:** $$ M.A = \frac{\text{Load}}{\text{Effort}} = \frac{\text{Effort arm}}{\text{Load arm}} = 1 $$ **Velocity Ratio (V.R):** If the effort moves a distance 'd' and the load also moves a distance 'd' (in downward direction for effort, upward for load): $$ V.R = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} = \frac{d}{d} = 1 $$ **Efficiency ($\eta$):** $$ \eta = \frac{M.A}{V.R} = \frac{1}{1} = 1 \text{ or } 100\% $$ **Uses of single fixed pulley:** 1. Used to change the direction of applied force. 2. It is used to lift a load in upward direction by applying force (or effort) in downward direction. *Example:* It is used to lift water in a bucket from a well, or to lift sand, bricks, and other building materials in a basket to the top of a building during construction work. #### Single Movable Pulley In a single movable pulley, one end of the string passes over the pulley and is attached to a fixed support and the effort is applied at the free end of the string in the upward direction. The load is lifted as the block of the pulley moves up. The tension in the string is equal to the effort applied, so the total upward pull on the pulley is `2 × effort`. **Mechanical advantage of a single movable pulley:** $$ M.A = \frac{\text{Load}}{\text{Effort}} = \frac{2 \times \text{Effort}}{\text{Effort}} = 2 $$ Thus, in a single movable pulley, the load lift is two times the effort applied. **Velocity Ratio (V.R):** If the load is pulled up through a distance 'd', then the effort is pulled through a distance '2d'. $$ V.R = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} = \frac{2d}{d} = 2 $$ **Efficiency ($\eta$):** $$ \eta = \frac{M.A}{V.R} = \frac{2}{2} = 1 \text{ or } 100\% $$ A single movable pulley is an example of a Class II lever. #### Combination of a Single Fixed and Movable Pulley The free end of the string passing over the single movable pulley passes over the single fixed pulley, and the effort is applied in the downward direction. The single movable pulley doubles the effort. Therefore, Load = 2 × Effort. **Mechanical advantage:** $$ M.A = \frac{\text{Load}}{\text{Effort}} = \frac{2 \times \text{Effort}}{\text{Effort}} = 2 $$ **Velocity Ratio (V.R):** If the load moves up by 'd', the effort moves down by '2d'. $$ V.R = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} = \frac{2d}{d} = 2 $$ **Efficiency ($\eta$):** $$ \eta = \frac{M.A}{V.R} = \frac{2}{2} = 1 \text{ or } 100\% $$ #### Combination of Pulleys: Block and Tackle System It consists of two pulley blocks. One block of pulleys is fixed to a rigid support and the other block of pulleys is movable. A strong, light, and inextensible string passes around all pulleys of the system. Effort (E) is applied at the free end of the string. Tension (T) in each part of the string is the same and continuous and supports the load (L). **Two types of block and tackle systems:** 1. **Number of pulleys in the lower movable block (n) = Number of pulleys in the upper fixed block.** *This block and tackle system has 4 pulleys (2 in upper, 2 in lower).* In this configuration, Load (L) = 4 × Effort. **Mechanical advantage of 4 pulleys block and tackle system:** $$ M.A = \frac{\text{Load}}{\text{Effort}} = \frac{4 \times \text{Effort}}{\text{Effort}} = 4 $$ 2. **Number of pulleys in the lower movable block (n) = One less than the number of pulleys in the upper fixed block.** *This block and tackle system has 5 pulleys (3 in upper, 2 in lower).* In this configuration, Load (L) = 5 × Effort. **Mechanical advantage of 5 pulleys block and tackle system:** $$ M.A = \frac{\text{Load}}{\text{Effort}} = \frac{5 \times \text{Effort}}{\text{Effort}} = 5 $$ In general, if a block and tackle system consists of 'n' pulleys, then the mechanical advantage of the system is 'n'. **Velocity Ratio and Efficiency for Block and Tackle System:** If the load is moved up by 'd' in a block and tackle system having 'n' pulleys, then the effort is moved by 'nd'. $$ V.R = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} = \frac{nd}{d} = n $$ **Efficiency ($\eta$):** $$ \eta = \frac{M.A}{V.R} = \frac{n}{n} = 1 \text{ or } 100\% $$ **Assumptions for having 100% efficiency in case of block and tackle system:** 1. The block and tackle system is frictionless. 2. The weight of the lower movable block of the system is negligible as compared to the load attached to it. **Efficiency of a block and tackle system having n pulleys, if the weight of lower movable block is not negligible as compared to the load attached to it.** Let W = weight of lower movable block of block and tackle system. Let L = Load attached to the lower movable block. Let nE = effort applied. Then, L + W = nE. Or L = nE - W. **Mechanical advantage, M.A:** $$ M.A = \frac{\text{Load}}{\text{Effort}} = \frac{nE - W}{E} = n - \frac{W}{E} $$ **Velocity Ratio (V.R):** $$ V.R = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} = \frac{nd}{d} = n $$ **Efficiency ($\eta$):** $$ \eta = \frac{M.A}{V.R} = \frac{n - \frac{W}{E}}{n} = 1 - \frac{W}{nE}