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A child is pushing a merry-go-round. The angle through which the merry-go- round has turned varies with time according to θ(t)=γt+βt3, where γ= 0.400 rad/s and β= 0.0120 rad/s3. (a) Calculate the angular velocity of the merry-go-round as a function of time. (b) What is the initial value of the angular velocity? (c) Calculate the instantaneous value of the angular velocity ω$$z at t= 5.00 s and the average angular velocity ωavz for the time interval t= 0 to t= 5.00 s. Show that ωavz is not equal to the average of the instantaneous angular velocities at t= 0 and t= 5.00 s, and explain.

Short Answer

Expert verified
(a) ω(t)=0.400+3×0.0120×t2. (b) 0.400 rad/s. (c) Instantaneous: 1.300 rad/s, Average: 0.7 rad/s. These differ due to non-linear acceleration.

Step by step solution

01

Expression for Angular Velocity

The angular velocity ω(t) is the derivative of the angle θ(t) with respect to time. Thus, we have: ω(t)=dθdt=ddt(γt+βt3)Differentiating gives:ω(t)=γ+3βt2
02

Initial Angular Velocity

To find the initial angular velocity, substitute t=0 into the expression for ω(t):ω(0)=γ+3β(0)2=γ=0.400rad/s
03

Instantaneous Angular Velocity at t = 5 s

Substitute t=5.00s into the expression for ω(t) to find the instantaneous angular velocity:ω(5)=0.400+3(0.0120)(5)2Calculate:ω(5)=0.400+3×0.0120×25=0.400+0.900=1.300rad/s
04

Average Angular Velocity Over Interval

The average angular velocity ωavz over the interval from t=0 to t=5 is given by:ωavz=θ(5)θ(0)50Calculate θ(5):θ(5)=0.400×5+0.0120×(5)3=2.0+1.5=3.5radians Since θ(0)=0,ωavz=3.505=0.7rad/s
05

Compare Averages

The average of the instantaneous angular velocities at t=0 and t=5 is:ω(0)+ω(5)2=0.400+1.3002=0.850rad/sThis differs from ωavz=0.7rad/s because the average angular velocity is affected by the acceleration term βt3, which changes the rate of rotation non-linearly over time.

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Key Concepts

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

Understanding Rotational Kinematics
Rotational kinematics is the branch of physics that deals with the motion of objects that rotate or spin. Objects like merry-go-rounds rotate around an axis, and rotational kinematics helps describe this motion.
  • Key terms include angular position, angular velocity, and angular acceleration.
  • The equations and concepts are similar to those used in linear kinematics but are adapted for rotation.
By knowing the angular position of an object over time, we can calculate its angular velocity and acceleration, giving us a complete picture of its rotational motion.
Exploring Angular Displacement
Angular displacement refers to the change in the angle through which an object has rotated, measured in radians. It helps us understand how much rotation has occurred.For a merry-go-round, if the initial angle is zero, the angular displacement can be calculated using the function θ(t)=γt+βt3where γ and β are coefficients that represent the constant and timedependent components of the motion, respectively.
  • This equation integrates time to determine how the angle changes as the object turns.
  • Understanding angular displacement helps in predicting future positions and in calculating other dynamic factors like velocity and acceleration.
Instantaneous Velocity Clarified
Instantaneous velocity is the speed of an object at a specific moment in time. For rotational motion, this is called angular velocity and is denoted as ω(t).The angular velocity indicates how quickly the object is rotating at any given point, and can be derived by differentiating the expression for angular displacement:ω(t)=dθdt=γ+3βt2
  • This formula reveals that angular velocity changes with time due to the term 3βt2.
  • At t=5 seconds, for instance, this function gives us the precise rotational speed at that moment, helping gauge performance or synchronization in mechanical systems.
Calculating Average Velocity
Average angular velocity over a period gives a broad picture of how an object has been rotating, smoothing out its speed variations into a single value.This is calculated by the change in angular displacement over time:ωavz=θ(5)θ(0)5
  • This formula computes the average rate of rotation from start to finish of a given time span.
  • Crucially, this value can differ from the mean of instantaneous velocities at the start and end times due to the nature of non-uniform motion.
Understanding both instantaneous and average angular velocities allows us to analyze rotational motion in different contexts, from simple machinery to complex engineering systems.

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Most popular questions from this chapter

An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250rev/s and a constant angular acceleration of 0.900rev/s2. (a) Compute the angular velocity of the turntable after 0.200 s. (b) Through how many revolutions has the turntable spun in this time interval? (c) What is the tangential speed of a point on the rim of the turntable at t=0.200 s? (d) What is the magnitude of the resultant acceleration of a point on the rim at t=0.200 s?

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On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a CD player, the track is scanned at a constant linear speed of v= 1.25m/s. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the CD is played. (See Exercise 9.20.) Let's see what angular acceleration is required to keep v constant. The equation of a spiral is r(θ)=r0+βθ, where r0 is the radius of the spiral at θ= 0 and β is a constant. On a CD, r0 is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, β must be positive so that r increases as the disc turns and θ increases. (a) When the disc rotates through a small angle dθ, the distance scanned along the track is ds=rdθ. Using the above expression for r(θ), integrate ds to find the total distance s scanned along the track as a function of the total angle θ through which the disc has rotated. (b) Since the track is scanned at a constant linear speed v, the distance s found in part (a) is equal to vt. Use this to find θ as a function of time. There will be two solutions for θ ; choose the positive one, and explain why this is the solution to choose. (c) Use your expression for θ(t) to find the angular velocity ωz and the angular acceleration αz as functions of time. Is αz constant? (d) On a CD, the inner radius of the track is 25.0 mm, the track radius increases by 1.55 mm per revolution, and the playing time is 74.0 min. Find r0,β, and the total number of revolutions made during the playing time. (e) Using your results from parts (c) and (d), make graphs of ωz (in rad/s) versus t and αz (in rad/s2) versus t between t= 0 and t= 74.0 min. TheSpinning eel. American eels (Anguilla rostrata) are freshwater fish with long, slender bodies that we can treat as uniform cylinders 1.0 m long and 10 cm in diameter. An eel compensates for its small jaw and teeth by holding onto prey with its mouth and then rapidly spinning its body around its long axis to tear off a piece of flesh. Eels have been recorded to spin at up to 14 revolutions per second when feeding in this way. Although this feeding method is costly in terms of energy, it allows the eel to feed on larger prey than it otherwise could.

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