An amusement park ride consists of a rotating circular platform 8.26 m in diameter from which 10 kg seats are suspended at the end of 1.14 m massless chains. When the system rotates, the chains make an angle of 16.2 degrees with the vertical. The acceleration of gravity is 9.8 m/s^2.

1. What is the speed of each seat? If i did it right i got 4.95 m/s

2. If a child of mass 26.2 kg sits in a seat, what is the tension in the chain (for the same angle)?

If the chains make 16 deg to the vertical, then

v^2/r / g= tan 16, so

v= sqrt(rgtan16)

but r= 4.23+1.14Sin16

I don't get your v.

To find the speed of each seat, you can use the concept of centripetal acceleration. Centripetal acceleration is given by the formula:

a = v^2 / r

where:
a = acceleration
v = velocity
r = radius

1. Given the diameter of the circular platform (D = 8.26 m), we can calculate the radius (r = D/2 = 8.26/2 = 4.13 m).

The angle the chains make with the vertical (θ = 16.2 degrees) gives us the tangential acceleration of the seat. To find the acceleration in the vertical direction, we need to calculate the gravitational force acting on the seat. The gravitational force (Fg) is given by:

Fg = m * g

where:
m = mass of the seat (10 kg)
g = acceleration due to gravity (9.8 m/s^2).

Using this, we can find the vertical acceleration (ay) as:

ay = Fg / m

Now, we know that the vertical acceleration (ay) is equal to the centripetal acceleration (a) since the chains prevent the seat from falling. Therefore,

a = ay = v^2 / r

To find the speed (v) of each seat, we rearrange the equation:

v = √(a * r)

v = √(ay * r)

Substituting the values,

v = √(9.8 * 4.13)

v ≈ 4.94 m/s (rounded to two decimal places)

So, the speed of each seat is approximately 4.94 m/s.

2. To find the tension in the chain when a child of mass 26.2 kg sits in a seat, we need to consider the forces acting on the seat. There are two forces: the tension in the chain (T) and the gravitational force (Fg) acting downwards.

The vertical component of the force exerted by the seat is given by:

Fy = T * cos(θ)

The gravitational force acting downwards is given by:

Fg = m * g

where:
T = tension in the chain
θ = angle made by the chain with the vertical (16.2 degrees)
m = mass of the child (26.2 kg)
g = acceleration due to gravity (9.8 m/s^2).

The equation for the forces in the vertical direction is:

Fy = T * cos(θ) - Fg

Since the seat is not moving vertically, the net force in the vertical direction is zero. Therefore,

T * cos(θ) - Fg = 0

Solving for tension (T):

T = Fg / cos(θ)

Substituting the given values,

T = (26.2 * 9.8) / cos(16.2)

T ≈ 264.67 N (rounded to two decimal places)

So, the tension in the chain, when a child of mass 26.2 kg sits in a seat, is approximately 264.67 N.