A dad pushes a small hand-driven merry-go-round tangentially and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume that the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 560 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edges. Calculate the torque required to produce the acceleration, neglecting the frictional torque. What force is required at the edge?

Short Answer

Expert verified

The required torque at the edge is \(320\;{\rm{N}} \cdot {\rm{m}}\).

Step by step solution

01

Concepts

Torque is the product of the moment of inertia and the square of the distance. For this problem, first, find out the angular acceleration of the system and multiply it with the total moment of inertia of the system.

02

Given data

The initial angular velocity of the merry-go-round is \({\omega _1} = 0\).

The final angular velocity of the merry-go-round is \({\omega _2} = 15\;{\rm{rpm}} = \left( {15 \times 2\pi } \right)\;{\rm{rad/s}}\).

The time taken by the merry-go-round is \(\Delta t = 10.0\;{\rm{s}}\).

The radius of the merry-go-round is \(r = 2.5\;{\rm{m}}\).

The mass of the disk is \(M = 560\;{\rm{kg}}\).

The mass of each child is \(m = 25\;{\rm{kg}}\).

The children are at the edge of the merry-go-round.

You can assume the children as point masses at the edge of the disc.

Let \(\tau \) be the torque required to produce such acceleration.

03

Calculation of torque

The total moment of inertia of the merry-go-round and the child system is

\(I = \frac{1}{2}M{r^2} + 2\left( {m{r^2}} \right)\).

Now, the angular acceleration of the system is \(\alpha = \frac{{{\omega _2} - {\omega _1}}}{{\Delta t}}\).

So, the required torque is

\(\begin{align}\tau &= I\alpha \\ &= \left\{ {\frac{1}{2}M{r^2} + 2\left( {m{r^2}} \right)} \right\}\frac{{{\omega _2} - {\omega _1}}}{{\Delta t}}\\ &= \left\{ {\left( {\frac{1}{2} \times \left( {560\;{\rm{kg}}} \right) \times {{\left( {2.5\;{\rm{m}}} \right)}^2}} \right) + 2\left( {\left( {25\;{\rm{kg}}} \right) \times \times {{\left( {2.5\;{\rm{m}}} \right)}^2}} \right)} \right\}\frac{{\left( {\frac{{15 \times 2\pi }}{{60}}} \right)\;{\rm{rad/s}} - 0}}{{10.0\;{\rm{s}}}}\\ &= 323.81\;{\rm{N}} \cdot {\rm{m}} \approx 320\;{\rm{N}} \cdot {\rm{m}}\end{align}\).

Hence, the required torque at the edge is \(320\;{\rm{N}} \cdot {\rm{m}}\).

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!

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 large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance l, holding onto it, Fig. 8–64. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 g’s?

A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.380 kg. Calculate (a) its moment of inertia about its center and (b) the applied torque needed to accelerate it from rest to 1750 rpm in 5.00 s. Take into account a frictional torque that has been measured to slow down the wheel from 1500 rpm to rest in 55.0 s.

A spherical asteroid with radius\(r = 123\;{\rm{m}}\)and mass\(M = 2.25 \times {10^{10}}\;{\rm{kg}}\)rotates about an axis at four revolutions per day. A “tug” spaceship attaches itself to the asteroid’s south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the asteroid’s surface as shown in Fig. 8–65. If\(F = 285\;{\rm{N}}\)how long will it take the tug to rotate the asteroid’s axis of rotation through an angle of 5.0° by this method?

Two blocks, each of mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–43. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system when it is first released.

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