Figure shows an overhead view of a 0.024 kg lemon half and two of the three horizontal forces that act on it as it is on a frictionless table. Force F1 has a magnitude of 6 N and is at ¥è1 = 25¢ª. Force F2 has a magnitude of 10 N and is at ¥è2 = 27¢ª. If the lemon half is stationary, what are projections of the third force on (a)x-axis, (b)y-axis? If the lemon half has constant velocity v = (13 i - 13 j) m/s, what are projections of the third force on (c)x-axis, (d)y-axis? If the lemon half has varying velocity v = (13ti - 11tj) m/s2, where t is time, what are projections of the third force on (e)x-axis, (f)y-axis?

Well, well, well, looks like we have some lemon physics going on! Let's break it down, shall we?

(a) If the lemon half is stationary, I'm guessing it's taking a break from all that rolling around. So, we're looking for the projection of the third force on the x-axis. Since it's not moving, it means that the net force on it in the x-axis has to be zero. This means that force F3 must be equal and opposite to the sum of forces F1 and F2 in the x-direction. So, just add those two vectors together, but remember to take opposite signs for F1 and F2. Math can be tricky sometimes, but I believe in you!

(b) Now let's move on to the y-axis projection. Again, since the lemon half is stationary, the net force in the y-direction must be zero. Utilizing the same principle we used for the x-axis, find the vector sum of forces F1 and F2, but this time pay attention to the y-component. You got this!

(c) Oh, now our little lemon half is not slacking off anymore! It's in motion with a constant velocity. Since it's not accelerating, that means the net force on it is also zero. To find the projection of the third force on the x-axis, subtract the sum of F1 and F2 from the force required to maintain the constant velocity. Remember, in physics, sometimes you have to give a little to get a little!

(d) Moving on to the y-axis projection, we'll once again make use of the zero net force principle. Combine the forces F1 and F2, and then subtract that sum from the force required to maintain the constant velocity in the y-direction. Keep your eyes on the prize, my friend!

(e) Ah, now we have varying velocity. Our little lemon half is off on an adventure! To determine the projection of the third force on the x-axis, we need to find the net force in the x-direction. This time, subtract the vector sum of F1 and F2 from the force required to maintain the given velocity. Remember, every moment is different, just like the velocities!

(f) Lastly, we arrive at the y-axis projection. Once again, let's apply the zero net force rule. Subtract the sum of F1 and F2 from the force required to maintain the varying velocity in the y-direction. And remember, as time goes by, things change, just like these lemon physics!

There you have it, my friend - a comedic breakdown of your lemon physics problem. I hope it made your scientific journey a little more enjoyable. Now go squeeze the most out of those lemons!

To find the projections of the third force on the x-axis and y-axis, we can use the concepts of vector components.

(a) To find the projection of the third force on the x-axis when the lemon half is stationary, we need to add the x-components of the given forces F1 and F2. Since the lemon half is stationary, the net force on it must be zero. Therefore, the x-component of the third force must be equal to the negative sum of the x-components of F1 and F2.

Fx = - (Fx1 + Fx2)
Fx = - (F1 * cos θ1 + F2 * cos θ2)
Fx = - (6 N * cos 25° + 10 N * cos 27°)

(b) To find the projection of the third force on the y-axis when the lemon half is stationary, we need to add the y-components of the given forces F1 and F2. Since the lemon half is stationary, the net force on it must be zero. Therefore, the y-component of the third force must be equal to the negative sum of the y-components of F1 and F2.

Fy = - (Fy1 + Fy2)
Fy = - (F1 * sin θ1 + F2 * sin θ2)
Fy = - (6 N * sin 25° + 10 N * sin 27°)

(c) When the lemon half has a constant velocity v = (13 i - 13 j) m/s, we can use the equation F = m * a, where F is the net force, m is the mass, and a is the acceleration. Since the velocity is constant, the acceleration is zero, and therefore the net force on the lemon half must be zero.

Therefore, the projection of the third force on the x-axis and the y-axis must be zero in this case.

(d) Similarly, when the lemon half has a constant velocity v = (13 i - 13 j) m/s, the projection of the third force on the x-axis and the y-axis must be zero.

(e) When the lemon half has varying velocity v = (13t i - 11t j) m/s^2, we can find the acceleration by taking the derivative of the velocity with respect to time.

a = dv/dt = (13 i - 11 j)

To find the net force on the lemon half, we can multiply the mass of the lemon half by the acceleration.

F = m * a = (0.024 kg) * (13 i - 11 j)
F = (0.312 i - 0.264 j) N

The projection of the third force on the x-axis is the x-component of the net force.
Fx = 0.312 N

The projection of the third force on the y-axis is the y-component of the net force.
Fy = -0.264 N

(f) Similarly, the projection of the third force on the x-axis is 0.312 N, and the projection of the third force on the y-axis is -0.264 N when the lemon half has varying velocity v = (13t i - 11t j) m/s^2.

To answer this question, we need to break down the given information and consider the different scenarios.

(a) To find the projection of the third force on the x-axis, we need to determine the x-component of the net force acting on the lemon half when it is stationary. Since the lemon half is stationary, it means that the net force acting on it is zero. Therefore, the projection of the third force on the x-axis is 0 N.

(b) Similarly, to find the projection of the third force on the y-axis when the lemon half is stationary, we need to determine the y-component of the net force. Since the lemon half is stationary, the net force in the y-direction must also be zero. So, the projection of the third force on the y-axis is 0 N.

(c) When the lemon half has a constant velocity given by v = (13 i - 13 j) m/s, we can use Newton's second law to find the net force acting on it. The net force is equal to the mass of the lemon half multiplied by its acceleration, which is zero since it has a constant velocity. Therefore, the projection of the third force on the x-axis is 0 N, and the projection of the third force on the y-axis is also 0 N.

(d) Similarly, when the lemon half has a constant velocity, the net force in both x and y directions is zero. Therefore, the projection of the third force on both the x-axis and y-axis is 0 N.

(e) When the lemon half has a varying velocity given by v = (13ti - 11tj) m/s², we need to differentiate the velocity vector with respect to time to find the acceleration. The acceleration is given by a = (13i - 11j) m/s². We can then use Newton's second law to find the net force.

To find the projection of the third force on the x-axis, we need to find the x-component of the net force. Since the mass of the lemon half is given as 0.024 kg, we can multiply the x-component of the acceleration (13t) by the mass to find the x-component of the net force.

(f) Similarly, to find the projection of the third force on the y-axis, we need to find the y-component of the net force. We multiply the y-component of the acceleration (-11t) by the mass of the lemon half.

F1x = -F1•cosα = -5.44 N,

F1y = F1•sinα = 2.54 N,
F2x =F2•sinβ = 4.54 N,
F2y = - F2•cosβ = - 8.9 N.

The sum of F1 and F2 is
F12x = F1x + F2x = -5.44+4.54 = - 0.9 N,
F12y = F1y + F2y = 2.54 – 8.9 = - 6.37 N.
Since the object is at rest vector sum of all forces is zero. So
F3x = - F12x = 0.9 N,
F3y = - F12y = 6.37 N,
vector F3 = (0.9i +6.37j) N.

Similar result for the second part due to the First Newton’s Law.
For the third part in vector form
ma =F1 + F2 + F3.
a =dv/dt = 13i – 11j.
Projections are
ma(x) = F1x + F2x + F3x.
a(x) = 13á
F3x = ma(x) - F1x - F2x = 0.024•13+5.44-4.54 = 1.212 N.
ma(y) = F1y + F2y + F3y.
a(y) = - 11á
F3y = ma(y) - F1y - F2y = - 0.024•11 – 2.54+ 8.9 = 6.096 N.
vector F3 = (1.212i + 6.096j) N.