Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

(III) A block of mass $m$ slides along a horizontal surfacelubricated with a thick oil which provides a drag forceproportional to the square root of velocity:$$F_{\mathrm{D}}=-b v^{\frac{1}{2}}$$If $v=v_{0}$ at $t=0,$ determine $v$ and $x$ as functions oftime.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

$v_{0} t-\frac{{v_{0}^{\frac{3}{2}}b}}{2 m} t^{2}+\frac{b^{2}}{12 m^{2}} t^{3}$

Physics 101 Mechanics

Chapter 5

Using Newton's Laws: Friction, Circular Motion, Drag Forces

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Winnipeg

Lectures

02:34

In physics, a rigid body is an object that is not deformed by the stress of external forces. The term "rigid body" is used in the context of classical mechanics, where it refers to a body that has no degrees of freedom and is completely described by its position and the forces applied to it. A rigid body is a special case of a solid body, and is one type of spatial body. The term "rigid body" is also used in the context of continuum mechanics, where it refers to a solid body that is deformed by external forces, but does not change in volume. In continuum mechanics, a rigid body is a continuous body that has no internal degrees of freedom. The term "rigid body" is also used in the context of quantum mechanics, where it refers to a body that cannot be squeezed into a smaller volume without changing its shape.

02:21

In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

02:22

A metal block of mass m sl…

06:40

A mass m has speed vo at t…

11:20

A block $0.1 \mathrm{m}$ s…

03:19

A particle of mass $\mathr…

04:30

(III) A small block of mas…

01:52

A smooth body of mass M sl…

A smooth body of mass $M$ …

Moving through a liquid, a…

04:34

Moving through a liquid, …

02:19

A block of weight $W$ slid…

14:24

(II) An object moving vert…

A block of mass $m$ slides…

03:08

so to solve this problem, we heavily need calculus. So if you have forgotten howto, um do David, it's an integration. Please go and review it and then try to solve the problem. So first off, we finally we need to find the and X as functions of time. So to find V as a function off tying and as well as excess a function of time. Now we're given the drag force F D, which is equal to negative B fee to the bar half Now. From here, um, we can do a bunch of things. First of all, we know that FT is the drag force. Then we have this should be called the mass Times acceleration where x elation is DDT off the so we can use that relation to solve for velocity over here. So what we can right here is M times. Do you read? 18. And from here, if we did this, uh, m divinity and minus b the half, then we have m d v d t is able to negative b he have. And from there it's all for, um velocity. So we can actually take velocity on the left and put time on the right. So we have Levi off. The to the powerhouse is equal to negative. Be over m dt. Now we need to take integration were given that at time equals zero. So we need to take integration to get Fi. So in other words, integration off any quantity. D X is equal to X plus c and if we take some limits than we can get rid of this sea or consent. So that's how we're trying to solve a fee from this innovation and were given that at equals to zero. Our velocity is the zero and anti equals two time T velocity must be equal to V. So from there you have, we have to solve the integration. So let's get on to the next. Actually, we can do it here and then we can solve the other part on the next page. So, just to recap, how do the integration? We know that X and off the X integration is ex off and plus one divided by n plus one so we can use the same thing here instead, off end being positive will have n being negative half because we can, right this part as devi me to the bar minus half. So B to the bar minus half. Yeah, so then we can use this formula right here, and it's all for it. And if we do so we see that, um, we have toe be house minus to be 0/2 is equal to minus B over M. So that's when we take the limits. The way we saw the limits is that's a, um The limits in this integration over here is, uh, from X, not to x. So then this quantity right here will get rid of the constant and we'll have x minus X. Not so It's up earning limit minus the Lord. That's what we did here. We subtracted the lower limit from the upper limit. So that's that's how we conveyed in terms off in a function of time. So if we actually modify this a question a little bit, we see that the is equal to V 0/2 minus deity Bye to end squared. Let's make it a smaller here. There we go. No, that's, uh, how we find the other function of time. Now, for the second part, we already mentioned that B X over DT is equal to tea or velocities the ex ability so we can substitute that station over here and we get d X by. Dt is equal to we not half minus e t over to m squared. Now again from here we can Katie on the right and solve the integration so we'll have the X, which is equal to the not half minus b t by two. M squared dt take the integration and for the limits at time equals zero x zero on a time equals sooty exes ex And from there with the similar argument, we see that X is equal to negative to end by three b be not to the power half minus BT by two m All right, cubed minus be not three way too. And ah, if we simplify that a bit, we have to end by three b sequel to be not three by two minus Be not half minus Reidy by two m Cute. Now we can go further and, uh, simplified this q factor and ah, then if we do so, we get eggs. Easy Will do toe Mbai three B's being on three by two minors be not three by two When St V not B t by two m plus she be not half be square T squared body M squared, minus beat. You teach you very eight and cute. From here we can write three m by three. Sorry to M by three b times V not toe the party If I do minus the knot to the bar t by do plus three v not Bt Very two m's minus she be not, uh be square t spread by m squared. Plus I beat you, teach you by eight and cute so we can get rid of this term And finally we get he not t minus the knock toe the party by do B bye toe em t squared plus me squared by 12 AM squared Do you cook? As you can see, it's a very long expression and Linda calculation, So make sure you do that by yourself and verify whether this expression is correct or not. Thank you

View More Answers From This Book

Find Another Textbook

Numerade Educator

Ffric Mkinctic X Foonn A box weighing 200 N is specd over a horizontal movcd…

04:53

Consider the objects on the coordinate grid: rod with m1 6.75 kg, right tria…

01:27

ueslion05 ptsA 7Og bullet moving east at 40 m/s strikes a 1.2kg bloc…

01:57

block of mass M = 2.50 kg is pushed 2.90 along a frictionless horizontal tab…

03:17

Calculate the time period for each of the following frequencies. 200 kHz 60 …

04:17

SerPSE1O 9.4 OP.024.My NotesAsk Your TeacherThe mass of Venus is…

01:11

SerPSE1O 9.9.0P.033.My NotesAsk Your Teacherrocket in space cons…

03:04

2 4cI4loading car is at rest on an inclined track The gross weight of th…

02:23

TS(ne centripetal acceleration in multiples of g of this point at full speed…