Work and Machines Workbook Pages

The Scientific Meaning and Calculation of Work

• Definition of Work: In scientific terms, work is performed only when a force is exerted on an object that causes that object to move a certain distance.

• Criteria for Work:     * A force must be applied to the object.     * The object must move a distance as a direct result of that force.     * The force must be exerted in the same direction as the object's motion.

• Examples of Work vs. No Work:     * Work: Pulling books out of a book bag; lifting a bin of newspapers; pulling a sled through the snow.     * No Work: Pushing on a car stuck in the snow (no motion occurs); holding a heavy piece of wood in place (no motion occurs); holding a bag of groceries (no motion occurs).

• Directionality of Force:     * Constant Velocity Carry: When carrying an object at a constant velocity, no work is being done on the object in the direction of motion. This is because the force exerted is upward (vertical) to support the weight, while the motion is horizontal. Since the force and motion are not in the same direction, work is zero.     * Pulling at an Angle: When pulling a sled by a rope at an angle to the ground, only the horizontal component of the force performs work because it is the only part of the force acting in the direction of the sled's motion.

• Factors Influencing Work:     * The amount of work depends on the amount of force exerted and the distance the object moves.     * Lifting a heavier object requires greater force than lifting a lighter object, thus resulting in more work for the same distance.     * Moving an object a shorter distance requires less work than moving it a greater distance, assuming force remains constant.

• Formula for Work:     * $\text{Work} = \text{Force} \times \text{Distance}$

• Units of Measurement:     * The SI unit of work is the Joule ($J$).     * One Joule ($1\,J$) is defined as the amount of work done when a force of $1\,Newton$ ($1\,N$) is exerted to move an object a distance of $1\,meter$ ($1\,m$).

Principles of Machines: Mechanical Advantage and Efficiency

• Definition of a Machine: A machine is a device that allows work to be performed in a more effective or easier way.

• Work Conservation: A machine does not decrease the total amount of work needed for a job; it simply changes how the work is done.

• Functions of a Machine: A machine makes work easier in three possible ways:     1. Multiplying the force exerted (input force).     2. Multiplying the distance over which the force is exerted.     3. Changing the direction in which the force is exerted.

• Force Definitions:     * Input Force: The force you exert on a machine.     * Output Force: The force exerted by the machine on an object.

• Mechanical Advantage (MA):     * Mechanical advantage is the number of times a force exerted on a machine is multiplied by the machine.     * Formula: $\text{Mechanical advantage} = \frac{\text{Output force}}{\text{Input force}}$     * If MA > 1, the output force is greater than the input force (force is multiplied).     * If MA < 1, the output force is less than the input force (distance is multiplied).

• Efficiency:     * In real-world applications, some work is always wasted to overcome friction.     * Efficiency compares the output work to the input work expressed as a percentage.     * Formula: $\text{Efficiency} = \frac{\text{Output work}}{\text{Input work}} \times 100\%$

• Mechanical Advantage Types:     * Actual Mechanical Advantage: The mechanical advantage a machine provides in a real situation, accounting for friction.     * Ideal Mechanical Advantage (IMA): The mechanical advantage of a machine in the absence of friction.

The Six Simple Machines

• Inclined Plane:     * Definition: A flat, slanted surface that allows the input force to be exerted over a longer distance.     * Formula: $\text{Ideal mechanical advantage} = \frac{\text{Length of incline}}{\text{Height of incline}}$     * Characteristics: The input force required is less than the output force, but exerted over a longer distance (e.g., a ramp).     * Efficiency can be increased by decreasing friction.

• Wedge:     * Definition: A device that is thick at one end and tapers to a thin edge at the other end.     * In a wedge, the inclined plane itself moves to multiply force (unlike a stationary ramp).

• Screw:     * Definition: An inclined plane wrapped around a cylinder.     * The spiral inclined plane forms the threads of the screw.     * Mechanism: When using a screwdriver, the input force is applied to the handle, which exerts a force on the threads. As the screw turns, the threads exert an output force on the material (e.g., wood).

• Lever:     * Definition: A rigid bar that is free to pivot or rotate around a fixed point called a fulcrum.     * Formula: $\text{Ideal mechanical advantage} = \frac{\text{Distance from fulcrum to input force}}{\text{Distance from fulcrum to output force}}$     * Classes of Levers:         * First-class: The fulcrum is located between the input force and the output force (e.g., seesaw, scissors, pliers).         * Second-class: The output force is located between the fulcrum and the input force (e.g., door, wheelbarrow, bottle opener).         * Third-class: The input force is located between the fulcrum and the output force (e.g., baseball bat, shovel, rake).

• Wheel and Axle:     * Definition: A simple machine made of two circular or cylindrical objects fastened together that rotate around a common axis.     * Formula: $\text{Ideal mechanical advantage} = \frac{\text{Radius of wheel}}{\text{Radius of axle}}$

• Pulley:     * Definition: A grooved wheel with a rope, chain, or cable wrapped around it.     * Fixed Pulley: Changes the direction of the input force but does not change the amount of force applied ($IMA = 1$).     * Movable Pulley: Has an ideal mechanical advantage of $2$.

Complex Systems and Human Applications

• Compound Machines: A machine that utilizes two or more simple machines. To calculate the mechanical advantage of a compound machine, the mechanical advantage of each individual simple machine involved must be known.

• Gears: A system of toothed wheels that fit into one another to transmit force and motion.

• Living Levers (Body Systems):     * Most levers in the human body consist of bones and muscles.     * Tendons: Tough connective tissue that attaches muscles to bones.     * Fulcrum: In the body, the joint near where the tendon is attached to the bone acts as the fulcrum.

• Working Wedges in the Body:     * Incisors: The front teeth resemble wedges. When biting, the wedge shape produces enough force to break items (like an ax splitting a log).

Mathematical Application Problems

• Work Calculation:     1. Work = $10\,N \times 35\,m = 350\,J$     2. Elevator lift: $500\,N$ weight lifted $30\,m$. $\text{Work} = 500\,N \times 30\,m = 15,000\,J$

• Mechanical Advantage Calculation:     3. $\text{Mechanical advantage} = \frac{60\,N}{15\,N} = 4$     4. Ramp usage: Lifting a desk requires $2,800\,N$ directly, but only $1,400\,N$ using a ramp. $\text{Mechanical advantage} = \frac{2,800\,N}{1,400\,N} = 2$

• Efficiency Calculation:     5. $\text{Efficiency} = \frac{100\,J}{200\,J} \times 100\% = 50\%$     6. Sledge hammer: Input work = $4,000\,J$; Output work on spike = $3,000\,J$. $\text{Efficiency} = \frac{3,000\,J}{4,000\,J} \times 100\% = 75\%$

• Inclined Plane IMA:     7. ${IMA} = \frac{8\,m}{2\,m} = 4$     8. Post office ramp: Length = $15\,m$, Height = $3\,m$. ${IMA} = \frac{15\,m}{3\,m} = 5$

• Lever IMA:     9. ${IMA} = \frac{4\,m}{2\,m} = 2$     10. Wheelbarrow: Handle distance (input) = $2.4\,m$, Stone distance (output) = $1.2\,m$. ${IMA} = \frac{2.4\,m}{1.2\,m} = 2$

• Wheel and Axle IMA:     11. ${IMA} = \frac{36\,cm}{3\,cm} = 12$     12. Bicycle wheel: Wheel radius = $30\,cm$, Axle radius = $5\,cm$. ${IMA} = \frac{30\,cm}{5\,cm} = 6$