An amusement park ride consists of a rotating
circular platform 8.09 m in diameter from
which 10 kg seats are suspended at the end
of 2.19 m mass less chains. When the system
rotates, the chains make an angle of 14.8
◦ with
the vertical.
The acceleration of gravity is 9.8 m/s
2
.
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To find the acceleration experienced by the seats on the amusement park ride, we need to consider the forces acting on the seats.
When the system rotates, the seats follow a circular path. This circular motion is caused by the tension force in the chains, which provides the centripetal force required for circular motion. The weight of the seats also acts vertically downward.
Using trigonometry, we can analyze the forces acting on the seats. Let's break down the forces:
1. Tension force (T) in the chains: The tension force acts radially inward towards the center of the circular platform. This force provides the centripetal force required to keep the seats in circular motion. The magnitude of the tension force in each chain can be found using the following equation:
T = m * a_c
where m is the mass of the seats and a_c is the centripetal acceleration.
2. Weight force (W) of the seats: The weight force acts vertically downward. Its magnitude can be calculated using the following equation:
W = m * g
where m is the mass of the seats and g is the acceleration due to gravity.
By considering the vertical component of the tension force and the weight force, we can determine the net force acting on the seats. This net force will be responsible for providing the centripetal acceleration required for circular motion. The net force can be found using the following equation:
Net Force = T * sinθ - W
where θ is the angle made by the chains with the vertical.
Since we have the diameter of the circular platform, we can calculate the radius by dividing the diameter by 2:
r = 8.09 m / 2 = 4.045 m
The centripetal acceleration can be found using the following equation:
a_c = ω^2 * r
where ω is the angular velocity. However, the value of ω is not given in the question. So, we need more information to find the centripetal acceleration and consequently the tension force in the chains and the net force acting on the seats.
If you have more information or if you can provide the angular velocity (ω) or the time taken for one revolution, I can help you calculate the acceleration experienced by the seats.
r = 2.19 sin 14.8 = .559 meters radius
centripetal a = Ac = omega^2 r =v^2/r
Look at tension in wire now
tan 14.8 = mg/mAc = 9.8/Ac
so
Ac = 9.8/tan 14.8 = 37.1 m/s^2
so v^2/r = 37.1
v^2 = 37.1* .559 = 20.7
v = 4.55 m/s
or for angular rate omega
omega^2 r = 37.1
omega^2 = 66.4
omega = 8.14 radians/ second