A large fake cookie sliding on a horizontal surface is attached to one end of a horizontal spring with spring constant k = 430 N/m; the other end of the spring is fixed in place. The cookie has a kinetic energy of 21.0 J as it passes through the spring's equilibrium position. As the cookie slides, a frictional force of magnitude 7.00 N acts on it. (a) How far will the cookie slide from the equilibrium position before coming momentarily to rest? (b) What will be the kinetic energy of the cookie as it slides back through the equilibrium position?
To solve this problem, we can use the principle of conservation of mechanical energy. According to this principle, the sum of the kinetic energy and potential energy of an object remains constant as long as no external work is done on the object.
Let's break down the problem into two parts:
(a) How far will the cookie slide from the equilibrium position before coming momentarily to rest?
In this part, we need to find the maximum distance the cookie will move before it comes to rest. Since the cookie is attached to a spring, it will experience a restoring force due to the spring.
The potential energy stored in a spring is given by the equation:
PE = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
Given that the cookie initially has kinetic energy (KE) of 21.0 J, we can equate the initial kinetic energy to the potential energy at the equilibrium position:
KE = PE
21.0 J = (1/2)(430 N/m)(0)^2
Since the potential energy at the equilibrium position is zero, we can solve for x:
0 = (1/2)(430 N/m)(x)^2
0 = 215x^2
This implies that x = 0 m since multiplying any number by zero gives zero. Therefore, the cookie will come to rest at the equilibrium position.
(b) What will be the kinetic energy of the cookie as it slides back through the equilibrium position?
Since the cookie initially has kinetic energy and comes to rest at the equilibrium position, the remaining energy (potential energy) will be converted into kinetic energy as the cookie slides back.
To find the potential energy when the cookie is at its maximum distance from the equilibrium position, we need to calculate the maximum displacement (displacement when the cookie comes momentarily to rest). Let's denote this maximum displacement as xmax.
Using conservation of mechanical energy again:
PE at xmax = KE initial
(1/2)k(xmax)^2 = 21.0 J
Rearranging the equation, we find:
(xmax)^2 = (2 * 21.0 J) / k
(xmax)^2 = (2 * 21.0 J) / 430 N/m
(xmax)^2 = 0.0977 m^2
Taking the square root on both sides, we get:
xmax = 0.312 m (approximately)
Therefore, the cookie will slide a distance of approximately 0.312 m from the equilibrium position before coming momentarily to rest.
To answer part (b), as the cookie slides back through the equilibrium position, it will have the same amount of kinetic energy as it had initially (21.0 J), since energy is conserved.
So, the kinetic energy of the cookie as it slides back through the equilibrium position will also be 21.0 J.