An ice cube of mass M is placed at rest in on a circular track of radius R, as shown at right. The position of the ice cube along the track follows that of a circle, given by: x^2 + (R – y)^2 = R^2 .
A)With respect to the bottom of the track, calculate the potential energy of the ice cube in terms of as a function of x.
B)Ignoring friction, the force governing the motion of the ice cube is conservative (it is part of gravity). Find the force acting on the ice cube at any position x. (Hint: recall relation between a conservative force and its potential function).
C)Show that for special case of x << R, the ice cube will execute simple harmonic motion. What is the restoring force?
D)What is the period of small oscillations of the ice cube as it rocks back and forth near the bottom of the bowl?
height = y so U = potential energy = M g y
so we need y as a function of x
(R-y)^2 = R^2-x^2
R-y = +/- (R^2-x^2)^.5
y = R +/- (R^2-x^2)^.5
for x < R and bottom half of the circle
y = R - (R^2-x^2)^.5
U = M g [ R - (R^2-x^2)^.5 ]
F = -m g dU/dx
F = -Mg d/dx [ R - (R^2-x^2)^.5 ]
= -Mg[- (R^2-x^2)^.5 ]
= Mg .5(R^2-x^2)^-.5 (-2 x)
= - Mg x /sqrt(R^2-x^2)
for x<<R
F = -Mg x/R which is a lot like a mass on a spring (F = -k x)
m a = -Mg x/R
if x = A sin wt
a = -w^2 x
M (-w^2 x ) = -Mg x/R
w^2 = (2 pi f)^2 = g/R
2 pi f = sqrt (g/R)
f = (1/2pi) sqrt (g/R)
T = 1/f = 2 pi sqrt (R/g)
y = R
To answer these questions, we need to use concepts from potential energy, conservative force, simple harmonic motion, and the period of oscillations. Here's how we can approach each part of the question:
A) To calculate the potential energy of the ice cube as a function of x, we can use the equation provided: x^2 + (R – y)^2 = R^2. Rearranging this equation, we get y = R - √(R^2 - x^2). The potential energy is given by the formula PE = mgh, where m is the mass of the ice cube, g is the acceleration due to gravity, and h is the height of the ice cube above the reference point (in this case, the bottom of the track). In this scenario, h = R - y, so the potential energy can be written as PE = mg(R - y). Substituting y = R - √(R^2 - x^2), we get PE = mg[R - (R - √(R^2 - x^2))]. Simplifying this expression will give us the potential energy as a function of x.
B) Since the force acting on the ice cube is conservative, we can find it as the derivative of the potential energy function with respect to x. In this case, the force acting on the ice cube is the negative gradient of the potential energy function. Taking the derivative of the potential energy function from part A with respect to x will give us the force as a function of x.
C) To show that the ice cube will execute simple harmonic motion for x << R, we need to demonstrate that the force acting on the ice cube is proportional to the displacement, and in the opposite direction. In other words, we need to show that the force follows Hooke's law: F = -kx, where k is the spring constant. By examining the equation for the force obtained in part B, we can determine if it adheres to Hooke's law for small x values.
D) The period of small oscillations for a mass-spring system can be calculated using the formula T = 2π√(m/k), where T is the period, m is the mass of the ice cube, and k is the spring constant. By comparing the equation for the force obtained in part B with Hooke's law, we can determine the equivalent spring constant (k) for the ice cube's motion. Substituting this value into the formula for the period will give us the answer to part D.
By following these steps, we can derive the answers to each part of the question.