an amusement park ride consists of a rotating circular platform 7.4 m in diameter from which 10 kg seats are suspended at the end of 1.81 m massless chains. when the system rotates, the chains make an angle of 26.6 degree with the vertical. the acceleration of gravity is 9.8 m/s^2.

a). what is the speed of each seat?
answer in units of m/s
b). if a child of mass 49.1 kg sits in a seat, what is the tension in the chain ( for the same angle)? answer in units of N

Where do I find the answers to this?

To solve this problem, we can use the principles of circular motion and the concepts of centripetal acceleration and tension in a system.

a) To find the speed of each seat, we need to find the centripetal acceleration. The centripetal acceleration (ac) is given by the formula:

ac = v^2 / r

where v is the velocity and r is the radius or half the diameter of the circular platform.

Given:
Diameter of the circular platform = 7.4 m
Radius (r) = 7.4 m / 2 = 3.7 m
Acceleration due to gravity (g) = 9.8 m/s^2

To find the velocity, we need to find the centripetal acceleration first. The centripetal acceleration can be calculated using the angle made by the chains with the vertical (θ). The vertical component of the tension in the chain provides the centripetal force. The vertical component of tension (T) is given by:

T = mg + m * ac * tan(θ)

where m is the mass of the seat and ac is the centripetal acceleration.

Given:
Mass of the seat (m) = 10 kg
Angle made by the chains with the vertical (θ) = 26.6 degrees = 26.6 * π / 180 radians

Using the formula above, we can calculate the centripetal acceleration (ac) by rearranging the formula as:

ac = (T - mg) / (m * tan(θ))

Substituting the given values:

ac = (T - (m * g)) / (m * tan(θ))
= (T - (10 kg * 9.8 m/s^2)) / (10 kg * tan(26.6 * π / 180))

Now, we can substitute the obtained value of ac into the earlier formula to find the velocity (v):

v = √(ac * r)
= √(ac * 3.7 m)

Calculating the value of v will give you the speed of each seat.

b) To find the tension in the chain when a child of mass 49.1 kg sits in the seat, we can use the same formula as before, but substitute the new mass (m) into it:

T = (m * g) + (m * ac * tan(θ))

Substituting the given values:

T = (49.1 kg * 9.8 m/s^2) + (49.1 kg * ac * tan(26.6 * π / 180))

Calculating the value of T will give you the tension in the chain.

Remember to always convert angles from degrees to radians when using trigonometric functions in mathematical calculations.