Vertical circular motion in uniform circular motion

In summary, a pilot is flying an airplane in a vertical circular loop with a radius of 1200 m. The speed of the plane is measured to be 150 m/s at the bottom of the circle where the total acceleration is 2.3g. The tangential acceleration of the plane is 12.5 m/s2 and the direction of the plane's acceleration at the bottom is 56.3 degrees. The correct formula for total acceleration is a = √(an^2 + at^2).
  • #1
seanpk92
6
0

Homework Statement


A pilot flies an airplane in a vertical circular loop with a radius of R = 1200 m. The plane is gaining speed as the pilot makes a dive, and its speed is measured to be 150 m/s when the plane reaches the bottom of the circle. If the total acceleration of the plane at the bottom is 2.3g, find (a) the tangential acceleration of the plane, (b) the direction of the plane’s acceleration at the bottom of the loop.


Homework Equations


What I can think of doing, is:
an=v2/R
and
a = an + at

The Attempt at a Solution


I tried making a free body diagram with the total force and an pointing toward the center, and the v and the at point to the right. At the bottom of the circle.
an = 1502/1200 = 18.75m/s2
a = 2.3g = 2.3/9.8m/s2 = 0.2346m/s2

but this doesn't give me my answer. The answer is (a) 12.5 m/s2 and (b) beta = 56.3o

(i don't know how to do a, so I can't start b)
 
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  • #2
seanpk92 said:

Homework Statement


A pilot flies an airplane in a vertical circular loop with a radius of R = 1200 m. The plane is gaining speed as the pilot makes a dive, and its speed is measured to be 150 m/s when the plane reaches the bottom of the circle. If the total acceleration of the plane at the bottom is 2.3g, find (a) the tangential acceleration of the plane, (b) the direction of the plane’s acceleration at the bottom of the loop.


Homework Equations


What I can think of doing, is:
an=v2/R
yes, good
and
a = an + at
the tangential and centripetal acceleration vectors are at right angles to each other, so you need to add them vectorially to get the total acceleartaion.

The Attempt at a Solution


I tried making a free body diagram with the total force and an pointing toward the center,
yes
]and the v and the at point to the right.
yes, and the tangential force points in the same direction
At the bottom of the circle.
an = 1502/1200 = 18.75m/s2
a = 2.3g = 2.3/9.8m/s2 = 0.2346m/s2
2.3 g's means the acceleration is 2.3 times the gravitational acceleration
but this doesn't give me my answer. The answer is (a) 12.5 m/s2 and (b) beta = 56.3o

(i don't know how to do a, so I can't start b)
Correct you formula for total acceleration
 
  • #3
Thank you so much PhantomJay!
 
  • #4
seanpk92 said:
Thank you so much PhantomJay!
(You're) Welcome to PF!:smile:
 
  • #5


I would like to clarify that the equations used in your attempt at a solution are not applicable for vertical circular motion. In uniform circular motion, the velocity and acceleration are always perpendicular to each other, but in vertical circular motion, the velocity and acceleration can have a component in the same direction. This means that the equations you used, namely a_n = v^2/R and a = a_n + a_t, are not valid for this scenario.

To solve this problem, we need to use the concept of centripetal acceleration, which is the acceleration towards the center of the circle. In this case, the total acceleration at the bottom of the loop is given as 2.3g, which includes both the tangential and centripetal components.

To find the tangential acceleration, we can use the equation a_t = v^2/R, where v is the speed at the bottom of the loop. Substituting the given values, we get a_t = (150 m/s)^2/1200 m = 18.75 m/s^2.

To find the centripetal acceleration, we can use the equation a_c = v^2/R, where v is the speed at the bottom of the loop. Substituting the given values, we get a_c = (150 m/s)^2/1200 m = 18.75 m/s^2.

Now, to find the direction of the acceleration, we can use the fact that the total acceleration is towards the center of the circle, i.e. it is in the downward direction. This means that the tangential acceleration, which is in the same direction as the velocity, must also be in the downward direction. Therefore, the direction of the plane's acceleration at the bottom of the loop is in the direction of the plane's velocity, which is downward.

To find the angle beta, we can use the inverse tangent function, i.e. beta = tan^-1(a_t/a_c). Substituting the values, we get beta = tan^-1(18.75/18.75) = 45 degrees (since a_t = a_c in this case).

Therefore, the answers are (a) 18.75 m/s^2 and (b) beta = 45 degrees.
 

What is vertical circular motion in uniform circular motion?

Vertical circular motion in uniform circular motion is a type of motion where an object moves in a circular path while maintaining a constant speed. The path of the object is perpendicular to the ground, and the object experiences a constant force towards the center of the circle.

What is the difference between vertical circular motion and horizontal circular motion?

The main difference between vertical circular motion and horizontal circular motion is the direction of the object's path. In vertical circular motion, the object moves in a path perpendicular to the ground, while in horizontal circular motion, the object moves in a path parallel to the ground. Additionally, the forces acting on the object may differ depending on the direction of the motion.

What is the centripetal force in vertical circular motion?

The centripetal force in vertical circular motion is the force that acts on an object towards the center of the circle, keeping the object in its circular path. This force is responsible for changing the direction of the object's velocity, but not its speed.

How does the radius affect the speed of an object in vertical circular motion?

The radius of the circular path affects the speed of an object in vertical circular motion. The smaller the radius, the greater the speed of the object, and vice versa. This is because the centripetal force is directly proportional to the square of the speed and inversely proportional to the radius.

What is the relationship between the velocity and acceleration in vertical circular motion?

In vertical circular motion, the velocity and acceleration are always perpendicular to each other. The velocity is tangent to the circular path, while the acceleration is directed towards the center of the circle. This means that the velocity and acceleration are constantly changing in direction, but the speed remains constant.

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