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Chapter 4: Problem 65

A purse at radius \(2.00 \mathrm{~m}\) and a wallet at radius \(3.00 \mathrm{~m}\) travel in uniform circular motion on the floor of a merry-go-round as the ride turns. They are on the same radial line. At one instant, the acceleration of the purse is \(\left(2.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\). At that instant and in unit-vector notation, what is the acceleration of the wallet?

Short Answer

Expert verified
The acceleration of the wallet is \((3.00 \mathrm{~m/s^2}) \hat{i} + (6.00 \mathrm{~m/s^2}) \hat{j}\).

Step by step solution

01

Understanding Circular Motion

In uniform circular motion, the acceleration vector is always directed towards the center, which means it will always have a centripetal component that can be calculated based on the radius of the circle.
02

Using the Given Acceleration

The given acceleration of the purse is \( \vec{a}_{purse} = (2.00 \mathrm{~m/s^2}) \hat{i} + (4.00 \mathrm{~m/s^2}) \hat{j} \). Both components represent centripetal acceleration in uniform circular motion.
03

Identifying Relationship of Radii

The purse is located at a radius of 2.00 m, and the wallet is located at a radius of 3.00 m. The centripetal acceleration is proportional to the radius, thus \( a_{wallet} = (r_{wallet} / r_{purse}) \times a_{purse}\).
04

Calculating Acceleration of Wallet

For each component: \( a_{wallet, \hat{i}} = (3.00 \mathrm{~m} / 2.00 \mathrm{~m}) \times (2.00 \mathrm{~m/s^2}) = 3.00 \mathrm{~m/s^2} \), and \( a_{wallet, \hat{j}} = (3.00 \mathrm{~m} / 2.00 \mathrm{~m}) \times (4.00 \mathrm{~m/s^2}) = 6.00 \mathrm{~m/s^2} \).
05

Writing the Final Answer in Vector Notation

Therefore, the acceleration of the wallet in unit vector notation is \( \vec{a}_{wallet} = (3.00 \mathrm{~m/s^2}) \hat{i} + (6.00 \mathrm{~m/s^2}) \hat{j} \).

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Key Concepts

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

Centripetal Acceleration
Centripetal acceleration is a crucial concept in understanding circular motion. It's the acceleration that keeps an object moving in a circle, directing it towards the center of the circle. This acceleration occurs even if the object keeps a constant speed, because it continuously changes direction.
Think of a car turning around a circular track. Even though its speed is constant, its direction changes, which means there is acceleration at play. This is the centripetal, or "center-seeking," acceleration.
  • Centripetal acceleration formula: \[ a_c = \frac{v^2}{r} \]where \( v \) is the tangential speed and \( r \) is the radius of the circle.
  • It is always perpendicular to the velocity vector of the object and points towards the center of the circle.
Centripetal acceleration enables us to predict how objects in circular paths behave, like the purse and wallet on a merry-go-round.
Uniform Circular Motion
Uniform circular motion refers to the motion of an object traveling at a consistent speed along a circular path. Despite the constant speed, an object in uniform circular motion undergoes continuous change in direction, resulting in centripetal acceleration.
In the case of a merry-go-round, both the purse and wallet follow a path like this. Though they might have different radii, the principles remain the same.
  • The speed, though constant, does not affect the object's constant centripetal acceleration, which focuses only on direction change.
  • This kind of motion highlights an essential balance between inertia (tendency to move straight) and the centripetal force pulling the object towards the center.
This concept helps us calculate things like how fast or slowly the merry-go-round can spin without letting the purse or wallet fly off.
Vector Notation
Vector notation provides a clear and precise way to describe physical quantities that have both magnitude and direction. In physics, acceleration, velocity, and force are often represented using vectors.
In circular motion problems, like our purse and wallet, it's common to express acceleration in vector notation. For example, \[ \vec{a}_{purse} = (2.00 \, \mathrm{m/s^2}) \, \hat{i} + (4.00 \, \mathrm{m/s^2}) \, \hat{j} \]
  • The \( \hat{i} \) and \( \hat{j} \) are unit vectors, indicating direction in a 2D plane, usually corresponding to the x and y axes.
  • By calculating each component of acceleration separately, it simplifies complex equations and gives exact results.
This form provides a comprehensive picture of how the vector quantity acts in multiple directions, essential for precise calculations in physics.

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Most popular questions from this chapter

A plane flies \(483 \mathrm{~km}\) east from city \(A\) to city \(B\) in \(48.0 \mathrm{~min}\) and then \(966 \mathrm{~km}\) south from city \(B\) to city \(C\) in \(1.50 \mathrm{~h}\). For the total trip. what are the (a) magnitude and (b) direction of the plane's displacement, the (c) magnitude and (d) direction of its average velocity, and (e) its average speed?

Music is still available on vinyl records that are played on turntables. Such a record rotates with a period of \(1.8 \mathrm{~s}\). For a record with a radius of \(16 \mathrm{~cm}\), find the centripetal accceleration of a point on the edge of the record.

A watermelon seed has the following coordinates: \(x=-5.0 \mathrm{~m}\), \(y=9.0 \mathrm{~m}\), and \(z=0 \mathrm{~m}\). Find its position vector (a) in unit- vector notation and as (b) a magnitude and (c) an angle relative to the positive direction of the \(x\) axis. (d) Sketch the vector on a right-handed coordinate system. If the seed is moved to the \(x y z\) coordinates \((3.00 \mathrm{~m}\), \(0 \mathrm{~m}, 0 \mathrm{~m}\) ), what is its displacement (e) in unit-vector notation and as (f) a magnitude and \((\mathrm{g})\) an angle relative to the positive \(x\) direction?

A cameraman on a pickup truck is traveling westward at \(20 \mathrm{~km} / \mathrm{h}\) while he records a cheetah that is moving westward \(30 \mathrm{~km} / \mathrm{h}\) faster than the truck. Suddenly, the cheetah stops, turns, and then runs at \(45 \mathrm{~km} / \mathrm{h}\) eastward, as measured by a suddenly nervous crew member who stands alongside the cheetah's path. The change in the animal's velocity takes \(2.0 \mathrm{~s}\). What are the (a) magnitude and (b) direction of the animal's acceleration according to the cameraman and the (c) magnitude and (d) direction according to the nervous crew member?

The position vector for an electron is \(\vec{r}=(6.0 \mathrm{~m}) \hat{\mathrm{i}}-\) \((4.0 \mathrm{~m}) \hat{\mathrm{j}}+(3.0 \mathrm{~m}) \hat{\mathrm{k}}\). (a) Find the magnitude of \(\vec{r}\). (b) Sketch the vector on a right-handed coordinate system.
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