An amusement park ride consists of a rotating
circular platform 9.87 m in diameter from
which 10 kg seats are suspended at the end
of 2.59 m massless chains. When the system
rotates, the chains make an angle of 39.3
◦ with the vertical. The acceleration of gravity is 9.8 m/s2.
what is the speed of each seat? Answer in m/s
radius of platform = 9.87/2 = 4.935 meters
r = radius of seat circle = 4.95 + 2.59 sin 39.3 = 6.59 meters
call seat mass m, never mind the 10 kg. The angle would not change even if a big kid were in it :)
m g down
m v^2/r horizontal
sin 39.3 = (v^2/r) / g
v^2 = r g sin 39.3
To determine the speed of each seat on the amusement park ride, we need to analyze the forces acting on the seats. Let's break down the problem step by step:
1. First, let's calculate the length of the chain from the center of the circular platform to the seat. This can be done using the Pythagorean theorem:
length of chain = √((radius of platform)^2 + (length of chain)^2)
Given the diameter of the platform is 9.87 m, and the length of the chain is 2.59 m, we can plug in these values:
length of chain = √((9.87/2)^2 + 2.59^2)
= √(4.935^2 + 6.7081)
= √(24.351225 + 6.7081)
= √31.059325
≈ 5.576 m
2. Next, let's find the vertical component of the force acting on each seat. This force is due to the weight of the seat and can be calculated as:
vertical force = mass of seat * acceleration due to gravity
Given the mass of the seat is 10 kg and the acceleration due to gravity is 9.8 m/s^2, we can substitute these values:
vertical force = 10 kg * 9.8 m/s^2
= 98 N
3. Now, we can determine the net force acting on each seat in the vertical direction. This net force is the vertical component of the tension in the chain, which can be calculated as:
net force = vertical force / cosine(angle)
Given the angle between the chain and the vertical is 39.3°, we can calculate the cosine of this angle and substitute this value:
net force = 98 N / cos(39.3°)
= 98 N / cos(0.6843)
= 98 N / 0.7659
≈ 127.948 N
4. Finally, we can find the speed of each seat by using the centripetal force. The centripetal force is provided by the tension in the chain and can be calculated as:
centripetal force = mass of seat * (velocity of seat)^2 / (length of chain)
We know that the net force acting on the seat (tension in the chain) equals the centripetal force:
centripetal force = net force = 127.948 N
Rearranging the equation, we get:
(velocity of seat)^2 = (net force * length of chain) / mass of seat
Plugging in the values:
(velocity of seat)^2 = (127.948 N * 5.576 m) / 10 kg
= 711.509 Nm / 10 kg
≈ 71.151 m^2/s^2
Taking the square root of both sides to solve for the velocity:
velocity of seat = √(71.151 m^2/s^2)
≈ 8.433 m/s
Therefore, the speed of each seat on the amusement park ride is approximately 8.433 m/s.