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 3.04 m/s, and an 65.2-kg person feels a 290-N force pressing against his back. What is the radius of a chamber?
To find the radius of the chamber, we can use the concept of centripetal force.
Step 1: Determine the centripetal force acting on the person:
The centripetal force is the force that keeps an object moving in a circular path. In this case, it is the force pressing against the person's back. Given that the person feels a force of 290 N, we can denote this as the centripetal force (Fc) acting on the person.
Fc = 290 N
Step 2: Calculate the acceleration of the person:
The centripetal force can be calculated using the formula:
Fc = m * a
where:
Fc = Centripetal force
m = Mass of the person
a = Acceleration
Rearranging the formula, we get:
a = Fc / m
Substituting the given values, we have:
Acceleration (a) = 290 N / 65.2 kg
Step 3: Calculate the radius of the chamber:
The acceleration can be calculated using the formula:
a = v² / r
where:
a = Acceleration
v = Velocity
r = Radius
Rearranging the formula, we get:
r = v² / a
Substituting the given values, we have:
Radius (r) = (3.04 m/s)² / Calculated acceleration from Step 2
After calculating the acceleration in Step 2, substitute its value into the formula to find the radius.
To find the radius of the chamber, we can use the principles of circular motion and centripetal force.
First, let's recall the formula for centripetal force:
F = m * a
Where:
F is the centripetal force
m is the mass of the object in motion
a is the acceleration of the object in motion
In this case, the centripetal force is the force pressing against the person's back, which is given as 290 N. The person's mass is 65.2 kg.
Now, let's find the acceleration using the formula for circular motion:
a = (v^2) / r
Where:
a is the acceleration
v is the velocity
r is the radius of the circle
In this case, the velocity is given as 3.04 m/s. We can rearrange the formula to solve for the radius:
r = (v^2) / a
Now, let's substitute the given values:
r = (3.04^2) / (290 / 65.2)
Calculating this expression will give us the radius of the chamber.