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Work Done by a Constant Force
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Today, we're going to discuss the definition of work in physics. Work is done when a force moves an object across a distance. Can someone tell me what the formula for work is?
Is it W equals force times displacement?
Close! The formula is W = F ยท ฮr, where F is the force applied and ฮr is the displacement. Remember, W stands for work, and it's a scalar quantity.
What does it mean when you say work is a scalar quantity?
Great question! Unlike vector quantities like force, which have both magnitude and direction, work only has magnitude. This leads us to consider the angle between the force and displacement. If the angle is 0 to 90 degrees, work is positive. Can anyone explain what happens when the angle is 90 degrees?
That's when the work done is zero, right?
Exactly! If the force is perpendicular to the direction of motion, no work is done. So, keep in mind these anglesโuse the mnemonic 'Work is Done When You Run' for easy recall.
What about when the force opposes motion?
Good point! Thatโs when work is negative, which occurs between angles of 90 and 180 degrees. Essentially, if the force acts against the displacement, it reduces the object's energy.
In summary, we've learned that work depends on both the force applied, the displacement, and their angle. Remember, work can be positive, negative, or zero based on these factors.
Work Done by a Variable Force
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Now, letโs discuss work done by a variable force. When forces change along a path, how can we calculate work done?
Do we just use the same formula as before?
Not quite! For variable forces, we look at infinitesimal work, dW. The formula is dW = F(x) ยท dr, meaning we need to integrate to find total work.
Could you give us an example where weโd use integration?
Of course! A common example is the work done by a spring, which follows Hooke's Law: F(x) = -kx. Here, we can set up the integral for work done from an initial to a final displacement.
So the total work done by a spring is W = โซ(-kx) dx?
That's right! When you integrate that from x_initial to x_final, you find the work W_spring = -1/2 k (x_final^2 - x_initial^2).
This sounds a bit complicated, but I can see how it works now. Itโs all about understanding what forces are acting.
Exactly! Understanding the forces is crucial. Itโs also helpful to visualize it graphically. The area under the force vs. displacement graph gives us the work done.
To summarize, for variable forces, we use integration to calculate total work, often seen with systems like springs. Knowing the relationship between force and displacement is key.
Applications of Work
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Let's connect work to energy. How does the work done on an object relate to its energy?
I think itโs like if you do work on an object, you give it energy, right?
Correct! The work-energy theorem tells us that the work done on an object equals its change in kinetic energy: W_net = ฮK.
Does this apply to potential energy too?
Great question! Absolutely, work done against gravity or any conservative force changes potential energy as well. We often see this in raised objects.
Can we have an equation for potential energy like we do for kinetic?
Yes! Near Earth's surface, this is U_g = mgh. When an object is lifted, work must be done against gravity, which converts to gravitational potential energy.
So both kinetic and potential energy are forms of energy connected by work done?
Exactly! Work is the bridge between forces and energy forms. Remember: energy can be transformed, but the total energy remains conserved in a closed system.
In conclusion, we have seen how work relates to energy, emphasizing the critical role work plays in both kinetic and potential energy changes.
Introduction & Overview
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Quick Overview
Standard
The section elaborates on the definition of work in physics, describing it as the scalar product of force and displacement. It explains under which conditions work is positive, negative, or zero, and introduces work done by variable forces, emphasizing the importance of this concept in relating forces to energy changes.
Detailed
In the physics context, work (W) is defined as the measure of energy transfer that happens when an object is moved over a distance by an external force. Mathematically, work is expressed as the scalar product of the force applied () and displacement (), denoted as W = 3 . The angle (BC) between the force and displacement vectors is crucial in determining the nature of work done. Positive work occurs when the angle is between 0 and 90 degrees, indicating the force contributes to the displacement. Negative work occurs when the angle is between 90 and 180 degrees, indicating that the force opposes the displacement. Zero work arises when the angle is 90 degrees, meaning the force is perpendicular to the motion. The section also discusses the calculation of work done by variable forces and includes an example related to spring forces, reinforcing the understanding of work in energy transfer contexts.
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Work Done by a Constant Force
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If a constant force Fโ\vec{F}F acts on an object and the object moves a displacement ฮrโ\Delta \vec{r}ฮr, the work WWW done by the force is the scalar product of force and displacement:
W=Fโโ
ฮrโ=F ฮr cos ฯ, W = \vec{F} \cdot \Delta \vec{r} = F\,\Delta r\,\cos\phi,
W=Fโ
ฮr=Fฮrcosฯ,
where ฯ\phiฯ is the angle between Fโ\vec{F}F and ฮrโ\Delta \vec{r}ฮr.
Detailed Explanation
In physics, work is defined as the energy transferred to or from an object through force acting over a distance. When a force moves an object through a distance, we calculate work using the formula: W = F * ฮr * cos(ฯ). Here, W is the work done, F is the magnitude of the force, ฮr is the magnitude of the displacement, and ฯ is the angle between the force and the direction of the displacement.
- If the force is in the same direction as the displacement (ฯ = 0ยฐ), all the force does work on the object, resulting in positive work.
- If the force is opposite to the direction of movement (ฯ = 180ยฐ), the work is negative, meaning the force is acting against the movement.
- If the force is perpendicular to the displacement (ฯ = 90ยฐ), no work is done because the force does not contribute to displacement in that direction.
Examples & Analogies
Imagine pushing a car on a flat road. If you apply a force while pushing straight in the direction the car is supposed to move, you do positive work on it. In contrast, if you push against the car while it is rolling downhill, youโre doing negative work since you're trying to stop its motion. Finally, if you apply a force vertically while the car moves horizontally, you donโt contribute to its motion in the direction it is moving, resulting in zero work done.
Types of Work Based on Angle
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Positive work when 0โโคฯ<90โ0^\circ \le \phi < 90^\circ0โโคฯ<90โ (force has a component along displacement).
Negative work when 90โ<ฯโค180โ90^\circ < \phi \le 180^\circ90โ<ฯโค180โ (force opposes motion).
Zero work if force is perpendicular to displacement (ฯ=90โ\phi = 90^\circฯ=90โ).
Detailed Explanation
The work done by a force can be characterized by the angle at which the force is applied relative to the direction of displacement.
- Positive Work: When the force has a component in the direction of displacement (0ยฐ โค ฯ < 90ยฐ), work is positive. This situation means that the force is aiding the motion.
- Negative Work: When the angle is between 90ยฐ and 180ยฐ (90ยฐ < ฯ โค 180ยฐ), the work done is negative, indicating that the force opposes the motion.
- Zero Work: If the force is applied at a right angle to the direction of motion (ฯ = 90ยฐ), no work is done since there is no displacement in the direction of the force.
Examples & Analogies
Think of carrying a heavy backpack while walking. The force of gravity is acting downwards (perpendicular to the direction you're walking), so even though you're exerting force, you're doing no work against the gravitational force while moving forward. However, if you were to lift the backpack upward as you walk, the work being done against gravity becomes positive. Conversely, if you were to push against the backpack backwards while walking, you would be doing negative work.
Work Done by a Variable Force
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If Fโ(x)\vec{F}(x)F(x) varies along the path, then the infinitesimal work dWdWdW is:
dW=Fโ(x)โ
drโ. dW = \vec{F}(x) \cdot d\vec{r}.dW=F(x)โ
dr.
Total work from xix_ixi to xfx_fxf:
W=โซxixfF(x) cos ฯ dx. W = \int_{x_i}^{x_f} F(x)\,\cos\phi\,dx.
W=โซxi xf F(x)cosฯdx.
Detailed Explanation
When a force acting on an object changes as the object moves, we can express the work done as an integral. The infinitesimal work, represented as dW, is calculated by the dot product of the varying force vector and the differential displacement vector: dW = F(x) โ dr. To find the total work done over a specific path where the force changes, we integrate this expression across the limits of the displacement from xi to xf, resulting in: W = โซ(xi to xf) F(x) cos(ฯ) dx. Here, we evaluate how the force applied at different points along the path affects the total work done.
Examples & Analogies
Consider compressing a spring. As you compress it, the force you need to apply increases due to Hooke's Law (F = -kx), where k is the spring constant. At the beginning of the compression, it may take less force, but as it gets compressed more, the required force increases. To find the total work done as you compress the spring from its relaxed position (xi) to the fully compressed position (xf), you would need to integrate the varying amount of force applied over the distance compressed.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Work is done when a force moves an object across a distance.
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Work can be positive, negative, or zero depending on the direction of the force related to displacement.
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Calculating work for variable forces often involves integration.
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The work-energy theorem connects work done by forces to changes in kinetic energy.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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A person pushing a box across the floor demonstrates positive work, as the force applied assists with displacement.
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When friction acts on a sliding object, it does negative work, reducing the object's kinetic energy.
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Lifting a weight against gravity involves doing positive work and converting this energy into gravitational potential energy.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
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For work done, itโs force and move, with angles right to make the groove.
๐ Fascinating Stories
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Imagine a runner pushing a boulder uphill. The more they push in the right direction (0ยฐ), the more they work! If they slip sideways (90ยฐ), their effort has no work.
๐ง Other Memory Gems
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Use the acronym 'POSITIVE' to remember that Positive Work occurs when Work is in the Same direction (angle 0ยฐ).
๐ฏ Super Acronyms
Remember 'W = Fยทฮr' stands for Work equals Force times Displacement.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Work
Definition:
The energy transferred when a force moves an object over a distance.
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Term: Scalar Product
Definition:
The product of two vectors that gives a scalar result, used in calculating work (W = Fยทฮr).
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Term: Displacement
Definition:
The change in position of an object, described as a vector quantity.
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Term: Force
Definition:
Any interaction that can change the motion of an object.
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Term: Positive Work
Definition:
Work done when the force has a component in the direction of displacement (0ยฐ โค ฯ < 90ยฐ).
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Term: Negative Work
Definition:
Work done when the force opposes the direction of displacement (90ยฐ < ฯ โค 180ยฐ).
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Term: Zero Work
Definition:
Work done when the angle between force and displacement is 90ยฐ, meaning force is perpendicular to motion.
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Term: Variable Force
Definition:
A force that changes in magnitude or direction along the path of motion.
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Term: Integrate
Definition:
To calculate the total work when dealing with variable forces by summation over infinitesimal increments.
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Term: Hooke's Law
Definition:
The principle stating that the force exerted by a spring is proportional to the displacement from its equilibrium position.