Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

A loaded grocery cart is rolling across a parking lot in a strong wind. You apply a constant force \(\overrightarrow{F} =(30 \, \mathrm{N})\hat{\imath} - (40 \, \mathrm{N})\hat{\jmath}\) to the cart as it undergoes a displacement \(\overrightarrow{s} = (-9.0 \, \mathrm{m})\hat{\imath} (3.0 \, \mathrm{m})\hat{\jmath}\). How much work does the force you apply do on the grocery cart?

Short Answer

Expert verified
The work done by the force on the cart is \(-390 \mathrm{J}\).

Step by step solution

01

Identify the Formula

The work done by a constant force is given by the dot product of the force vector \( \overrightarrow{F} \) and the displacement vector \( \overrightarrow{s} \). Mathematically, this is expressed as \( W = \overrightarrow{F} \cdot \overrightarrow{s} \).
02

Apply the Dot Product Formula

The dot product of two vectors \( \overrightarrow{A} = a_1 \hat{\imath} + a_2 \hat{\jmath} \) and \( \overrightarrow{B} = b_1 \hat{\imath} + b_2 \hat{\jmath} \) is calculated as \( \overrightarrow{A} \cdot \overrightarrow{B} = a_1b_1 + a_2b_2 \).
03

Substitute the Values

Substitute the given values of the force vector \( \overrightarrow{F} = (30 \, \mathrm{N})\hat{\imath} - (40 \, \mathrm{N})\hat{\jmath} \) and the displacement vector \( \overrightarrow{s} = (-9.0 \, \mathrm{m})\hat{\imath} + (3.0 \, \mathrm{m})\hat{\jmath} \) into the dot product formula. This means calculating \( W = 30(-9.0) + (-40)(3.0) \).
04

Calculate the Components

Calculate each component separately: - For the \( \hat{\imath} \) component: \( 30 \times -9.0 = -270 \)- For the \( \hat{\jmath} \) component: \((-40) \times 3.0 = -120 \).
05

Sum the Components

Add the two results obtained from the components: - \( -270 + (-120) = -390 \).
06

State the Result

The total work done by the force is \(-390 \mathrm{J}\). The negative sign indicates that the force applied is opposite to the direction of displacement.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Dot Product
The dot product is a mathematical tool used when dealing with vectors. It's particularly useful when we need to calculate work, as in the problem of the rolling grocery cart.
  • What is a dot product? The dot product of two vectors results in a scalar, which is a single number, not a vector.
  • How is it used? When you have two vectors expressed in a Cartesian coordinate system, such as \(\overrightarrow{F} = (a_1)\hat{\imath} + (a_2)\hat{\jmath}\) and \(\overrightarrow{s} = (b_1)\hat{\imath} + (b_2)\hat{\jmath}\), the dot product is calculated as \(a_1b_1 + a_2b_2\).
  • Why use it for work calculations? In physics, work is defined as the force applied in the direction of the displacement. The dot product naturally incorporates this directional component, providing the magnitude of work done by the force.
The grocery cart exercise showcases how the dot product simplifies calculations involving force and displacement. You multiply the corresponding components of two vectors and sum them up, yielding energy used or needed.
Force and Displacement in Work Calculations
Force and displacement are the heart of work and energy calculations. Understanding how they interact is vital when solving physics problems.
  • Force: This is any interaction that, when unopposed, changes the motion of an object. It's a vector quantity, meaning it has both a magnitude and a direction. In the exercise, the force vector is given as \(\overrightarrow{F} = (30 \, \mathrm{N})\hat{\imath} - (40 \, \mathrm{N})\hat{\jmath}\).
  • Displacement: This is the change in position of an object. Like force, it's also a vector. For the rolling cart, the displacement is \(\overrightarrow{s} = (-9.0 \, \mathrm{m})\hat{\imath} + (3.0 \, \mathrm{m})\hat{\jmath}\).
  • Relationship in Work: Work occurs when a force causes displacement. The amount of work done is determined by both the magnitude of the force and how much it contributes to moving an object along the direction of displacement.
Through understanding these vectors and their components, you'll realize how vital each part is in determining the net work done, as shown in the problem's step-by-step solution.
Mastering Vector Mathematics
Vector mathematics is essential for solving many physics problems, especially those involving forces and motions, like the grocery cart problem.
  • Vector Basics: A vector represents a quantity with both a magnitude and a direction. In mathematical terms, they are often expressed in components, using unit vectors such as \(\hat{\imath}\) and \(\hat{\jmath}\).
  • Components and Notation: Each vector component corresponds to a dimension (e.g., horizontal and vertical). For example, \(\overrightarrow{F} = (30 \, \mathrm{N})\hat{\imath} - (40 \, \mathrm{N})\hat{\jmath}\) shows force components along the x and y axes.
  • Mathematical Operations: Vectors can be added, subtracted, and multiplied. Multiplication includes the dot product, which is key in calculating work, as it takes into account the directional agreement between force and displacement.
Understanding vector operations and notation helps simplify complex physics problems. By practicing these operations, you'll find that tasks such as calculating work from vectors become much more manageable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A pump is required to lift 800 kg of water (about 210 gallons) per minute from a well 14.0 m deep and eject it with a speed of 18.0 m/s. (a) How much work is done per minute in lifting the water? (b) How much work is done in giving the water the kinetic energy it has when ejected? (c) What must be the power output of the pump?

When a car is hit from behind, its passengers undergo sudden forward acceleration, which can cause a severe neck injury known as \(whiplash\). During normal acceleration, the neck muscles play a large role in accelerating the head so that the bones are not injured. But during a very sudden acceleration, the muscles do not react immediately because they are flexible; most of the accelerating force is provided by the neck bones. Experiments have shown that these bones will fracture if they absorb more than 8.0 J of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 ms, what is the greatest speed this car and its driver can reach without breaking neck bones if the driver's head has a mass of 5.0 kg (which is about right for a 70-kg person)? Express your answer in m/s and in mi/h. (b) What is the acceleration of the passengers during the collision in part (a), and how large a force is acting to accelerate their heads? Express the acceleration in m/s\(^2\) and in \(g\)'s.

As part of your daily workout, you lie on your back and push with your feet against a platform attached to two stiff springs arranged side by side so that they are parallel to each other. When you push the platform, you compress the springs. You do 80.0 J of work when you compress the springs 0.200 m from their uncompressed length. (a) What magnitude of force must you apply to hold the platform in this position? (b) How much \(additional\) work must you do to move the platform 0.200 m \(farther\), and what maximum force must you apply?

You throw a 3.00-N rock vertically into the air from ground level. You observe that when it is 15.0 m above the ground, it is traveling at 25.0 m/s upward. Use the work\(-\)energy theorem to find (a) the rock's speed just as it left the ground and (b) its maximum height.

Three identical 8.50-kg masses are hung by three identical springs (\(\textbf{Fig. E6.35}\)). Each spring has a force constant of 7.80 kN/m and was 12.0 cm long before any masses were attached to it. (a) Draw a free-body diagram of each mass. (b) How long is each spring when hanging as shown? (\(Hint\): First isolate only the bottom mass. Then treat the bottom two masses as a system. Finally, treat all three masses as a system.)

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free