Select the correct answer.
A ball is whirled overhead at a velocity of 12 meters/second in a circular path whose radius is 2 meters. Select the value listed below that must be substituted for $c$ if the equation for determining centripetal force is defined in this way:
[tex]x=\frac{\mu_a}{a}[/tex]
A. 2
B. 5
C. 6
D. 12
Answer (1)
Relate the given equation to the centripetal force equation F = r m v 2 .
Identify that a corresponds to the radius r .
Substitute the given radius value, which is 2 meters, for a .
The value to be substituted for c (assuming a typo and it meant a ) is 2 .
Explanation
Understanding the Problem We are given that a ball is whirled overhead at a velocity of 12 meters/second in a circular path whose radius is 2 meters. The equation for determining centripetal force is defined as x = a μ a . We need to find the value that must be substituted for c .
Relating Variables to Centripetal Force In the context of centripetal force, the equation is usually given as F = r m v 2 , where F is the centripetal force, m is the mass, v is the velocity, and r is the radius. Comparing this with the given equation x = a μ a , we can make the following analogies:
x corresponds to the centripetal force F .
μ a corresponds to m v 2 .
a corresponds to the radius r .
We are given that the velocity v = 12 m/s and the radius r = 2 meters. We need to find the value that corresponds to c .
Identifying the Correct Value The problem states that we need to find the value that must be substituted for c . However, the variable c does not appear in the equation x = a μ a . It seems there is a typo in the problem. We are given the options A. 2, B. 5, C. 6, D. 12. Since a corresponds to the radius r , and the radius is given as 2 meters, the value to be substituted for a is 2. If we assume that the question meant to ask which value should be substituted for a , then the answer would be 2. Also, the velocity is 12 m/s, so if the question meant to ask which value should be substituted for the velocity v , then the answer would be 12. However, since the question asks for the value to be substituted for c , and c does not appear in the equation, we must assume that the question is asking for the value of a , which is the radius.
Final Answer Since a corresponds to the radius r , and the radius is given as 2 meters, the value to be substituted for a (or possibly c due to the typo) is 2. Therefore, the correct answer is A. 2.
Conclusion The value that must be substituted for c (assuming the question meant a ) is 2.
Examples
Understanding centripetal force is crucial in designing safe and effective amusement park rides. For example, when engineers design a roller coaster loop, they need to calculate the centripetal force acting on the cars to ensure that the passengers remain safely in their seats throughout the loop. By correctly substituting the velocity and radius of the loop into the centripetal force equation, they can determine the necessary forces and design the loop accordingly, providing a thrilling yet safe experience for riders. The equation F = r m v 2 helps engineers to calculate these forces accurately.
