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AnnZ [28]
3 years ago
10

Two identical objects, A and B, are sitting on a table. If the net force on object A is 5 N and the net force on object B is 10

N, which of the following statements is true?
A. Neither object will accelerate.
B. Object A will accelerate more quickly than object B.
C. Object B will accelerate more quickly than object A.
D. The acceleration of each object will depend on other forces than the net force.
Physics
1 answer:
sveta [45]3 years ago
7 0

If the net force on object A is 5 N and the net force on object B is 10 N, then object B will accelerate more quickly than object A provided the mass of both objects are same.

Answer: Option C

<u>Explanation: </u>

According to Newton’s second law of motion, any external force applied on an object is directly proportional to the mass and acceleration of the object. In order to state this law in terms of acceleration, it is stated that acceleration exhibited by any object is directly proportional to the net force applied on the object and inversely proportional to the mass of the object as shown below:

                      \text {Acceleration of the object } \propto \frac{\text {Net force on the object}}{\text {Mass of the object}}

So if two objects A and B are identical which means they have same mass, then the acceleration attained by the object will be directly proportionate to the net forces exerted on the objects only.

Thus if the force applied is more for one object, then the object will be exhibiting more acceleration compared to the other one. So as object B is experiencing a net force of 10 N which is greater than the net force experiences by object A, then the object B will be accelerating more quickly compared to the object A's acceleration.

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A projectile of mass m is launched with an initial velocity vector v i making an angle θ with the horizontal as shown below. The
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Angular momentum is given by the length of the arm to the object, multiplied by the momentum of the object, times the cosine of the angle that the momentum vector makes with the arm. From your illustration, that will be: 
<span>L = R * m * vi * cos(90 - theta) </span>

<span>cos(90 - theta) is just sin(theta) </span>
<span>and R is the distance the projectile traveled, which is vi^2 * sin(2*theta) / g </span>

<span>so, we have: L = vi^2 * sin(2*theta) * m * vi * sin(theta) / g </span>

<span>We can combine the two vi terms and get: </span>

<span>L = vi^3 * m * sin(theta) * sin(2*theta) / g </span>

<span>What's interesting is that angular momentum varies with the *cube* of the initial velocity. This is because, not only does increased velocity increase the translational momentum of the projectile, but it increase the *moment arm*, too. Also note that there might be a trig identity which lets you combine the two sin() terms, but nothing jumps out at me right at the moment. </span>

<span>Now, for the first part... </span>

<span>There are a few ways to attack this. Basically, you have to find the angle from the origin to the apogee (highest point) in the arc. Once we have that, we'll know what angle the momentum vector makes with the moment-arm because, at the apogee, we know that all of the motion is *horizontal*. </span>

<span>Okay, so let's get back to what we know: </span>

<span>L = d * m * v * cos(phi) </span>

<span>where d is the distance (length to the arm), m is mass, v is velocity, and phi is the angle the velocity vector makes with the arm. Let's take these one by one... </span>

<span>m is still m. </span>
<span>v is going to be the *hoizontal* component of the initial velocity (all the vertical component got eliminated by the acceleration of gravity). So, v = vi * cos(theta) </span>
<span>d is going to be half of our distance R in part two (because, ignoring friction, the path of the projectile is a perfect parabola). So, d = vi^2 * sin(2*theta) / 2g </span>

<span>That leaves us with phi, the angle the horizontal velocity vector makes with the moment arm. To find *that*, we need to know what the angle from the origin to the apogee is. We can find *that* by taking the arc-tangent of the slope, if we know that. Well, we know the "run" part of the slope (it's our "d" term), but not the rise. </span>

<span>The easy way to get the rise is by using conservation of energy. At the apogee, all of the *vertical* kinetic energy at the time of launch (1/2 * m * (vi * sin(theta))^2 ) has been turned into gravitational potential energy ( m * g * h ). Setting these equal, diving out the "m" and dividing "g" to the other side, we get: </span>

<span>h = 1/2 * (vi * sin(theta))^2 / g </span>

<span>So, there's the rise. So, our *slope* is rise/run, so </span>

<span>slope = [ 1/2 * (vi * sin(theta))^2 / g ] / [ vi^2 * sin(2*theta) / g ] </span>

<span>The "g"s cancel. Astoundingly the "vi"s cancel, too. So, we get: </span>

<span>slope = [ 1/2 * sin(theta)^2 ] / [ sin(2*theta) ] </span>

<span>(It's not too alarming that slope-at-apogee doesn't depend upon vi, since that only determines the "magnitude" of the arc, but not it's shape. Whether the overall flight of this thing is an inch or a mile, the arc "looks" the same). </span>

<span>Okay, so... using our double-angle trig identities, we know that sin(2*theta) = 2*sin(theta)*cos(theta), so... </span>

<span>slope = [ 1/2 * sin(theta)^2 ] / [ 2*sin(theta)*cos(theta) ] = tan(theta)/4 </span>

<span>Okay, so the *angle* (which I'll call "alpha") that this slope makes with the x-axis is just: arctan(slope), so... </span>

<span>alpha = arctan( tan(theta) / 4 ) </span>

<span>Alright... last bit. We need "phi", the angle the (now-horizontal) momentum vector makes with that slope. Draw it on paper and you'll see that phi = 180 - alpha </span>

<span>so, phi = 180 - arctan( tan(theta) / 4 ) </span>

<span>Now, we go back to our original formula and plug it ALL in... </span>

<span>L = d * m * v * cos(phi) </span>

<span>becomes... </span>

<span>L = [ vi^2 * sin(2*theta) / 2g ] * m * [ vi * cos(theta) ] * [ cos( 180 - arctan( tan(theta) / 4 ) ) ] </span>

<span>Now, cos(180 - something) = cos(something), so we can simplify a little bit... </span>

<span>L = [ vi^2 * sin(2*theta) / 2g ] * m * [ vi * cos(theta) ] * [ cos( arctan( tan(theta) / 4 ) ) ] </span>
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