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For the particle of Prob. $12.76,$ determine the tangential component $F_{t}$ of the central force $\mathbf{F}$ along the tangent to the path of the particle for $(a) \mathbf{u}=0,(b) \mathbf{u}=45^{\circ}$

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Physics 101 Mechanics

Chapter 12

Kinetics of Particles: Newton’s Second Law

Newton's Laws of Motion

University of Michigan - Ann Arbor

University of Sheffield

University of Winnipeg

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

09:37

Isaac Newton (4 January 1643 – 31 March 1727) was an English mathematician, physicist, astronomer, theologian, and author (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

08:28

For the particle of Prob. …

14:24

01:31

Ssuppose that a particle f…

01:43

A particle of mass $m$ is …

05:08

in this problem. We have a particle that is moving under a central force. And so the angular momentum per unit mass age is a constant. And if we use equation 12.27 we can see that age is also R squared. Deter dot And since this is a constant, we can call it h not, and h not is defined by are not the not the initial position times the initial velocity. So using that expression, we confined an expression for tita dot So, tita dot is simply are not the not over r squared. And so we can write this as are not the note and we can write out r squared as are not squared cost great theater and so one are not canceled from the top in the bottom, so that becomes the not over are not cost squared off data so that we have an expression for theater dot and will use this a bit later. So now we will calculate the radio component off the velocity. The radio component of velocity is VR and we are simply are dot So I'll not is the time derivative DDT off our, which is are not conspirator and from the change role this is minus are not signed pita peter dot So that's the radio component of the velocity In terms of pita. Now, the trans verse component of V, which is V pita is equal to are peter dot and using our expression for tita dot or using our expression for our rather we get this to be are not co sign of data simply times data dot So now we have both the radio and the transfers component of the velocity And so we confined the speed of the particle V So V, we know, is simply the square root off the squares the sum of squares off the radio and transfers components VR square plus the teeter squared on this simplifies to are not peter dot and now we can open up data dot So this becomes are not times we not all the are not course squared off data and now it becomes clear why we expand the data dot And so this is equal to see the Arnaud's cancel. So this is V note over costs cost squared pita. So we have an expression for the speed of the particle in terms off the angle, pita. Now that we have V, we can actually calculate the tangential component off the acceleration. Mhm. If we have a tendency to component acceleration, we can use that to calculate the tangential component of the force. So the conventional component of acceleration a T we know is D v D t. The rate of change of velocity. And so this is we not minus two, minus science. Tita data dot So we take our expression for V, and we differentiated with respect to time. So again, we'll have to use the chain rule. It's over cost great, Peter. So this is equal to to we not signed Pita over called squid Deaton multiplied by peter dot which is we not over are not costs grade data. So this simplifies to to the north squared scientific data over are not actually this is above. This is Cost Cube, Peter. So this becomes are not times cost data to the power five. So now we have the financial component off the acceleration. So since we have the financial computer deceleration, we confined the tangential component off the force. The financial component of the force, F T is equal to M 80 by Newton's second law. And so that's the mass off the particle, multiplied by the expression for the tangential force. So that's two m v, not squared science, Tita. All the are not cause enough data to the fourth power. So now we have an expression for the tangential force. So the tangential force we need to find into scenarios. So in part eight, data is equal to zero. If we let Tita equal to zero into the expression above, we get that obviously signed teachers equal to zero. And so this tangential force F T is equal to zero for part B. If we let Pita equal to 45 degrees and we substitute that into that expression, we get the tangential force. F T is to, um we not into sign 45 off are not into cost, Peter. The first power off details 45 45 degrees. And we get that the international force simplifies 28 mm. The not squared over are not

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