Three astronauts, propelled by jet backpacks, push on a 120...

Three astronauts, propelled by jet backpacks, push on a 120 kg asteroid exerting the forces shown with F1 = 42.0N, F2 = 75.0N, F3 = 51.0N, ?1 = 25.0°, and ?3 = 65.0° 1a. Find FR, the resultant force vector, that the astronauts are exerting on the asteroid. Use i, j or <FRx, FRy> form. 1b. With what magnitude of force and standard position angle would a fourth astronaut have to push on the asteroid in order to keep the asteroid in translational equilibrium? Give a magnitude ||F4|| and a standard position angle ?4.

Transcript

00:01 Hello student in this question we have given this is a asteroid and on this asteroid three astronauts are applying three astronauts are applying forces like this f1 f2 and f3 at an angle theta 1 and theta 3 now the question is asking find the resultant of this all forces so the first part we have to answer the resultant of all three forces that is the resultant of all three forces that is the forces applied by the astronaut we have to find the first part that is the result so in order to find the first part let's write the force f1 f2 and f3 in i cap and j kfm so force f1 we can write in i kp form like this f1 cos theta 1 i cap plus f1 sine theta 1 j cap right now let's write the next force f2 the force f2 we can write like this as the force f2 has only x component that is i component so it will be f2 i cap now let's write the force f3 the force f3 will be like this f3 cost theta 3 i cap minus f3 sine teta 3 j cap so we we wrote all the forces in icp jkf format now let's find the resultant of all these forces as we know the result of the vector sum of the vector sum of all these three forces now let's write right let's write each forces individually here in order to find the resultant so force f1 we can write as f1 cost theta 1 i cap plus f1 j cap then the force f2 will be f2 i cap and the force f3 will be f3 cos theta 3 i cap minus f3 sine teta 3 j cap now let's write i component and j component side by side let's group them so that we will easy to so it will be easy to solve for us we can write f1 cost theta 1 f2 plus f3 f3 this is the i part now let's write the j part f1 sine theta 1 minus f3 sine theta 2 which is is the j part.

03:27 Now let's substitute the values.

03:30 Now let's write the values of f1 and cost theta all that.

03:35 So it will be 40 cost 25 degree.

03:42 F2 we know it is a 75 newton.

03:45 F3 will be 51 cost 65 i -cap plus f -1 is 402 that and sign 25 degree minus 51 sign 65 degree this is g cap now in solving this part now substituting the value of cost 25 or 65 and then multiplying we will get our i component at 134 .62 i kip and the g cap we will get the component as 28 .4.

04:30 So the first part of the question was find the resultant of all these three forces which is applied by the astronaut so the resultant of all these three forces is this the second part is asking find the magnitude of force find the magnitude of that force who will make the asteroid in translational translational equilibrium so let's first understand what do you mean by translational equilibrium in translational equilibrium the net force acting on an object or a body will be zero here our body is asteroid and we know the force the net force acting on here we know our body is asteroid here we know our body is asteroid and the force acting on that asteroid is resultant effort so in order to be make this for 0 we must apply an equal and opposite force f this will make our force this will make our this resultant force to be 0 and we will be in translational equivalent so we can see from here the force f will be equal and opposite to resultant force and we know our result from this this is our resultant force we can write here like i cap minus 28 .47 jk.

06:15 Now taking negative sign negative sign inside we can write like this 134 .6 to i cap minus minus will become plus 28 .47 jk.

06:34 So this is our force which will make the resultant force to be zero...

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