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(III) An object of mass $m$ is constrained to move in a circle ofradius $r .$ Its tangential acceleration as a function of time is givenby $a_{\text { tan }}=b+c t^{2},$ where $b$ and $c$ are constants. If $v=v_{0}$ at$t=0,$ determine the tangential and radial components of theforce, $F_{\text { tan }}$ and $F_{\mathrm{R}},$ acting on the object at any time $t>0$

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$$\frac{m}{r}\left(v_{0}+b t+\frac{1}{3} c t^{3}\right)^{2}$$

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

University of Michigan - Ann Arbor

Hope College

University of Sheffield

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.

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A particle of mass $m$ uni…

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Let $a_{r}$ and $a_{t}$ re…

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A particle of mass $m$ is …

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Soto find the tangential force. It's simply the mass times the initial exploration where the initial acceleration is given us e t. It is being plus C T squared and ah, tendon Jill Force will be just mass times 80 which is m times B plus C t squared. Now, for the, um, radial component out for the radial force, we'll see that the reading forces related toe uh, m times e are or the radial component central. Relax, elation or the radial acceleration, which can be related toe the financial velocity, which is, uh, college vts detention velocity s Oh, this is gonna be m v t squared. Delighted we are. So that's actually right. This is smaller so that it's consistent. No, uh, we see that there's a way to relate this initial X elation with an initial velocity. And the way we do that is from calculus from calculus. We know that 80 is equal to DDT offi or, in other words, HVT is equal to integration off 80 times. Dt. So that means we need to do an integration over the potential exploration to get the velocity. So let's actually do that. In a separate page, and then we'll come back to ah, this page and Phyllis out. So as we mentioned it and then she'll is equal to B plus C t squared. No, From the rule of integration, we see that if we're integrating off, we're integrating x n Then this should give us X n plus one divided by N plus one as a result. So in our case, to find out Vt, we need to just simply integrate off integrate b plus c t squared over DD and then we can actually separate them out because we have a plus sign. So this becomes B d t plus integration off C P squared duty where the first term will be simply be times t because we see that b is a constant. So all we're doing is doing his indication of bt, which is nothing but teeth. That's what we wrote here. And for the second time, we can use this relation over here. So in our case, X S t and N is too. So that's gonna be a plus and is too, as we mentioned, so we'll have to plus one which is 1/3 then instead of X. We have tea. So Tito Tito, the power to plus one which is three. And then, as we mentioned as we see that he is a constant here so we can replace one by the constant See. And since it's an intubation, uh, actually, we should have ah, constant around here. So it's just a constant. So, uh, this whole integration, plus a constant term. Now we need to get rid of this constant because ah, it's it's just ah, a constant term which can be an ingot read off using the information that's provided here. So from the information that's provided, we see that at equals to zero. If we call that velocity as v zero Or in other words, we know that at any people to zero, the velocity is just the initial velocity. And we did not the initial velocity as zero. So what that means is, ah, from this expression, when time equal to zero. Then we put a dying clue zero over here. So the 1st 2 times are zero. Then we have just constant. And as we mentioned, that V as at equal zero is V zero. So the constant is basically these Europe now we can use that relation over here. So finally, this expression, because Beatty plus see over treaty killed plus V zero. All right, so now we have the expression for Vt. Let's use that here. So this expression will be em over our times be zero plus beatty for us. 1/3 CD cute. And then we put a square because we have a vt is quite over here. So that's how we got our radio. Uh, force and tangential force was easy. It's just m times 80. Thank you.

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