Child on a swing problem...?

Question

A 44.0 kg child sits in a swing supported by two chains, each 3.01 m long.

(a) If the tension in each chain at the lowest point is 357 N, find the child's speed at the lowest point.

(b) What is the force exerted by the seat on the child at the lowest point? (Neglect the mass of the seat.)

Since people think all I want is the answer, I'll post what I've done so far. I know that:

sum (F) = m*a and that centripetal acceleration = (v^2)/r.

Tension - Fw = m* (v^2)/r
Fw = mg = (44.0 kg)(9.8 m/s/s) = 431.2 N
m * (v^2/r) = (44.0 kg)(v^2/3.01 m)

I just really have no idea how to get the velocity of it. I know how to get the tension afterwards...

Answer 1

Okay, this one's easy.

a) The difference between the tension and child's weight is the force that gives the centripetal acceleration (and therefore the centripetal force).

Let Fc be centripetal force, Fw - weight of a child and T - tension of one chain, l - the lenght of chain, m - mass of child, v - speed, g - free fall acceleration

Fc = 2*T - Fw
(m*v^2)/l = 2*T - m*g (here we multiply both sides by l and divide by m and it equals)
v^2 = (2*T*l)/m - g*l (take the square root)
v = sqrt[(2*T*l)/m - g*l] = 4.4 (m/s)

b) Force exterted on the child is equal to child's weight at the lowest point. Let's call it normal force - N

N = m*(g-a), where a - centripetal acceleration.

N = m*(g-v^2/l) = m*(g - (2*T)/m + g) = 2*m*g - 2*T = 148 (N)

(I'm not really sure about the second answer, because I didn't understand the text 100% clear, but I think it should be right).

Answer 2

Key to working this problem is to recognize that the kid, swing, and chains (i.e., the system) are not moving downward or upward at the lowest point and in that instant. So the sum of all forces on that system has to be zero. Otherwise, the system would be accelerating and moving in one direction or the other straight up or down.

So you need to first ID all the forces acting straight up and down (at the low point in the swing):

Tension (t1 and t2) on the two chains.
Weight of the chains, which you apparently assumed to be zero.
Weight (W) of the kid.
Weight of the swing (apparently you assumed it to be zero).
Centripetal/centrifugal force (C) from the swinging.

Because the system is not accelerating up or down at its low point, tension = the weights + centripetal force = t1 + t2 = W + C = m(g + v^2/r) = 2(357) = 44(9.81 + v^2/3.01); so that, by rearranging the terms, you can solve for v, the tangential velocity (speed) at the lowest point. Recognize that v along the chains (vertically up or down) at that low point is zero.

Because the only forces acting upward are the tension forces, they are the only forces that offset the downward forces on the seat. So the force on the seat has to be the sum of the two tensions t1 + t2. Otherwise, the system would be acceleratiing and moving in an up or down vertical direction.

Answer 3

Do your homework!