8. At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of 3.60 m/s, and an 87.0-kg person feels a 590-N force pressing against his back. What is the radius of the chamber? m
Gravitational force
In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.
Acceleration Due to Gravity
In fundamental physics, gravity or gravitational force is the universal attractive force acting between all the matters that exist or exhibit. It is the weakest known force. Therefore no internal changes in an object occurs due to this force. On the other hand, it has control over the trajectories of bodies in the solar system and in the universe due to its vast scope and universal action. The free fall of objects on Earth and the motions of celestial bodies, according to Newton, are both determined by the same force. It was Newton who put forward that the moon is held by a strong attractive force exerted by the Earth which makes it revolve in a straight line. He was sure that this force is similar to the downward force which Earth exerts on all the objects on it.
![**Problem 8: Centripetal Force in an Amusement Park Ride**
At an amusement park, there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis with their backs against the outer wall. At one point, the outer wall moves at a speed of 3.60 m/s, and an 87.0-kg person feels a 590-N force pressing against their back. What is the radius of the chamber?
Input your answer: ___________ m
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### Explanation:
To solve this problem, we need to use the concept of centripetal force, which is the force that keeps an object moving in a circular path. The centripetal force \( F_{c} \) is given by the formula:
\[ F_{c} = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the object (87.0 kg in this case),
- \( v \) is the velocity of the object (3.60 m/s here),
- \( r \) is the radius of the circular path.
Rearrange the formula to solve for the radius \( r \):
\[ r = \frac{mv^2}{F_{c}} \]
Given:
- \( F_{c} = 590 \) N
- \( m = 87.0 \) kg
- \( v = 3.60 \) m/s
Substitute these values into the equation:
\[ r = \frac{(87.0 \, \text{kg}) \times (3.60 \, \text{m/s})^2}{590 \, \text{N}} \]
\[ r = \frac{87.0 \, \times \, 12.96}{590} \]
\[ r = \frac{1128.72}{590} \]
\[ r \approx 1.91 \, \text{m} \]
Therefore, the radius of the chamber is approximately 1.91 meters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd0a9c9c-b0c8-47bb-88dc-3ac08f167320%2Fb702acf1-06f1-4fac-b529-cf94bb5a07d8%2Fkvgifm_processed.jpeg&w=3840&q=75)
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