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The masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ slide on the smooth (frictionless)inclines fixed as shown in Fig. $55 .(a)$ Determine a formula forthe acceleration of the system in terms of $m_{\Delta}, m_{\mathrm{B}}, \theta_{\mathrm{A}}, \theta_{\mathrm{B}}$ and $g .(b)$ If $\theta_{\mathrm{A}}=32^{\circ}, \theta_{\mathrm{B}}=23^{\circ},$ and $m_{\mathrm{A}}=5.0 \mathrm{kg},$ whatvalue of $m_{\mathrm{B}}$ would keep the system at rest? What would be thetension in the cord (negligible mass) in this case? (c) Whatratio, $m_{\mathrm{A}} / m_{\mathrm{B}}$ , would allow the masses to move at constantspeed along their ramps in either direction?

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a. the expression for the acceleration of the block is $\left[\frac{\left(m_{B} \sin \theta_{B}-m_{A} \sin \theta_{A}\right) g}{\left(m_{A}+m_{B}\right)}\right]$b. Rounding off to two significant figures, the mass of the block B is 6.8 $\mathrm{kg}$ the tension in the cord is $[26 \mathrm{N}]$c. the ratio of two blocks is $[0.74]$

Physics 101 Mechanics

Chapter 4

Dynamics: Newton's Laws of Motion

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Moment, Impulse, and Collisions

Cornell University

University of Michigan - Ann Arbor

University of Winnipeg

McMaster University

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.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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(III) Two masses $m_A=$2.0…

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A block (mass $m_{A} )$ ly…

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(II) Two masses $m_{A}=2.0…

10:32

Figure $11-35$ shows two m…

03:58

Two masses are connected b…

03:23

(11) Figure 35 shows two m…

06:04

As shown in the figure, bl…

02:14

Consider the above configu…

00:57

03:46

(II) Two masses, $m_{\math…

So this would be the free body diagram for the system that we have here and for we're gonna apply Newton's second law to both masses. So fair, massive A we have The sum, of course, is in the ex direction. This is going to be equal to force tension minus m sub A g sign of fate US of a. Uh this is equaling M sub eight times A for the master B. We have the sum of forces in the X direction equaling AMSA be times g times sign of fate us that be this is gonna be minus force tension. This will equal m sub b times a And now we're gonna add these to a crate to equations Together we can say forced tension minus amps of a G sign of data sub A plus. I'm so be G sign of status of B minus force of tension. This would equal and some eight times a plus. M c b times a force. Tension, of course, cancels out and we find that the acceleration is gonna be equal to M C B time. Sign of status of B minus mass of a time sign of fate us up A This would all be divided by the total mess. So this would be our answer for part A Now, for the, uh except for part B for the acceleration to be at rest acceleration. So at rest this means that the acceleration must equal zero meters per second squared. Ah, therefore the acceleration would equal. I'm so be again. Sign of fate us A B minus mass of a sign of fate us of a divided by massive A plus massive p and I forgot for this part. Um, there's a factor of G. My apologies Just realized so. So this entire term would be times the acceleration due to gravity. Okay, so here and there will be times G. This would equal zero at this point if this equal zero This means that Massa be time sign of status that be I would be equal two massive a time sign of fate s of a therefore massive B would be equal to massive eh Times Sign of fate us of a divided by sign of fate a sub B This is gonna equal 5.0 kilograms. Uh, most applied by sign of 30 two degrees divided by sign of 23 degrees, and this is equaling 6.8 kilograms. So if the massive block A is equal in five kilograms and the massive block B's equaling 6.8 kilograms, the system will be at rest and acceleration would be equal to zero meters per second. If we wanted to find attention here, we can say that for the massive, eh? The force tension minus mass of a times G sign of status of A. This is equaling zero. And so the force tension would simply be equal to the massive A 5.0 kilograms multiplied by the acceleration due to gravity 9.80 meters per second squared times sign of 32 degrees and we find that the forest tension is gonna be 26 Newton's if the if the system is at rest and four part the sorry poor for for part c uh, just like in part be, the acceleration will be zero for a constant velocity. So we can say for constant velocity, the acceleration is again zero meters per second squared. Ah, therefore the ah, the same exact relationship applies. However, we can say that the massive a divided by the massive be then equals G sign of Fada, so be divided by G sign of fate. US of a, the G's cancel out and we have sign of 23 degrees divided by sign of 32 degrees. This is equal in 0.74 So that would be your answer for part C. That is the end of the solution. Thank you for watching.

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