Alan

(a).  Because the plane is rising at constant velocity (not accelerating) the net vertical force must be zero.  That is the upward force of 8000*sin(65) must equal the weight of the plane.

(b).  Forward acceleration is obtained from mass*acceleration = 8000*cos(65°).  The mass of the plane is its weight divided by g (acceleration of gravity).

(c).  There are two forces acting on the plane:  The 8000N in the direction shown and its weight acting vertically down.  From (a) you know the net vertical force is zero, so can you figure out what is left?

I'll start you off:

The relationships between current (i) and voltage (v) for capacitor and inductor are:

i = C*dv/dt     for capacitor

v = L*di/dt     for inductor

If they are connected in series with an ac source (V*sin(ω*t)) the following circuit equation can be written:

L*di/dt + v = V*sin(ω*t)  where v is the voltage across the capacitor.

Differentiate this with respect to time and use the capacitor equation above to get:

L*d²i/dt² + i/C = ω*V*cos(ω*t)

Solve this 2nd-order differential equation for i as a function of t, making use of your initial conditions.

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The slope is the coefficient of the x term, so slope = -3/2 here.

The y-intercept is the constant term, so y-intercept = -3 here.

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Note that density = mass/volume.

The volume occupied by the sodium is Vsodium = 90/970 L = 1000*90/970 mL or 92.78mL.

This reduces the volume available for the gas, so its new density = 4.6/(250 - 92.78) g/mL

$${\frac{{\mathtt{4.6}}}{\left({\mathtt{250}}{\mathtt{\,-\,}}{\mathtt{92.78}}\right)}} = {\frac{{\mathtt{230}}}{{\mathtt{7\,861}}}} = {\mathtt{0.029\: \!258\: \!364\: \!075\: \!817\: \!3}}$$

new density ≈ 0.029 g/mL

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Yes, that many Newtons.

$${\mathtt{45}}{\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{35}}^\circ\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{9.8}} = {\mathtt{74.810\: \!939\: \!635\: \!795}}$$

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By the way, don't confuse the symbol N for the normal force with the symbol N meaning Newtons!

N (normal force) = 75 N (Newtons)

Without the extra s in the denominator the mass is given by the uglier expression:

More explanation of the process is given in the answer at the link given above by Melody.

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If the (1+s)³ term is in the numerator, the density would rapidly increase as s increased and the total mass would be infinite (unless a maximum limit for the size of s were supplied), so I assumed it was in the denominator.

The volume, dV, of a "thin" spherical shell, of thickness, ds, is given by the surface area of a sphere of radius s, namely 4.pi.s², multiplied by the small thickness ds.  So dV = 4.pi.s².ds.  The mass per unit volume within this thin shell is 1/(s.(1+s)³), so the mass within this shell is [4.pi.s²/(s.(1+s)³)].ds.  The total mass of the galaxy is obtained by "summing" all these shell masses from zero to infinity.  In other words, we let the shell thickness tend to zero and integrate from zero to infinity.

Does that help?

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The force exerted by the block on the ground is F1*sin(35) + m*g  or  45*sin(35°) + 5*9.8 N.

From Newton's 3rd law of motion (action and reaction are equal and opposite) this is the normal force exerted on the block.

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You can write

width = height - 2

Rearrange this to get height =, then replace width by 2.

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