Question

An object of mass m makes n revolutions per second around a circle of radius r at a constant speed. What is the kinetic energy of the object?

answer choices
A. 0
B. 1/2 π^ 2mn^ 2r^ 2
C. 2π ^2mn ^2 r ^2
D. 4π^2mn ^2r ^2

Answer 1

The kinetic energy of an object with mass m making n revolutions per second around a circle with radius r at a constant speed is given by the formula 2π²mn²r², which is option C in the answer choices.

Step-by-step explanation:

To calculate the kinetic energy of an object with mass m making n revolutions per second around a circle with radius r at a constant speed, first, we need to determine the tangential velocity (v) of the object. By definition, velocity is the circumference of the circle (which is 2πr) times the number of revolutions per second (n), resulting in v = 2πrn. The formula for kinetic energy (K) is K = 1/2 mv². Substituting the expression for v we get: K = 1/2 m(2πrn)². After simplifying, the kinetic energy is K = 1/2 m4π²n²r², which equals 2π²mn²r². Therefore, the correct answer is option C: 2π²mn²r².

Answer 2

The kinetic energy of the object is D. 4π²mn²r².

Step-by-step explanation:

Circular motion: The object's motion in a circle is defined by its angular velocity (ω) in radians per second, which relates to its revolutions per second (n) by:

ω = 2πn

Linear velocity: The object's linear velocity (v) at any point on the circle can be calculated by:

v = ωr

Kinetic energy: The kinetic energy (KE) of the object is then:

KE = ½mv²

Substituting the expression for v:

KE = ½m(ωr)²

Replacing ω with 2πn:

KE = ½m(2πn)²r²

Simplifying the expression:

KE = 4π²mn²r²

Therefore, the correct answer is D. 4π²mn²r². This expression accounts for the object's mass, angular velocity, and the radius of its circular path, providing the accurate value for its kinetic energy.

Option D is answer.