A large fake cookie sliding on a horizontal surface is attached to one end of a horizontal spring with spring constant k = 375 N/m; the other end of the spring is fixed in place. The cookie has a kinetic energy of 20.0 J as it passes through the position where the spring is unstretched. As the cookie slides, a frictional force of magnitude 10.0 N acts on it.
(a) How far will the cookie slide from the position where the spring is unstretched before coming momentarily to rest?
m
(b) What will be the kinetic energy of the cookie as it slides back through the position where the spring is unstretched?
J(a) How far will the cookie slide from the position where the spring is unstretched before coming momentarily to rest?
m
(b) What will be the kinetic energy of the cookie as it slides back through the position where the spring is unstretched?
J
To find the distance the cookie will slide before coming to rest, we need to consider the work done on the cookie. The work done by the frictional force is equal to the change in kinetic energy.
(a) The work done by friction is given by the equation:
Work = Force * Distance
The frictional force is 10 N, and the work done is equal to the change in kinetic energy, which is 20 J. So we can set up the equation:
10 N * Distance = 20 J
Rearranging the equation, we find:
Distance = 20 J / 10 N
Distance = 2 m
Therefore, the cookie will slide 2 meters before coming to rest.
(b) To find the kinetic energy of the cookie as it slides back through the position where the spring is unstretched, we need to consider the conservation of mechanical energy. The total mechanical energy consists of the kinetic energy and the potential energy stored in the spring.
At the position where the spring is unstretched, the potential energy of the spring is zero, so the total mechanical energy is equal to the kinetic energy. We can calculate this by using the formula:
Total Mechanical Energy = 0.5 * k * x^2
Where k is the spring constant and x is the displacement of the spring. Since the spring is momentarily at rest, the total mechanical energy is equal to the final kinetic energy.
Plugging the given values into the equation, we have:
0.5 * 375 N/m * x^2 = 20 J
Rearranging the equation to solve for x:
x^2 = (20 J) / (0.5 * 375 N/m)
x^2 = 0.1067 m^2
x = 0.326 m (rounded to three decimal places)
Therefore, the distance the cookie will slide back through the position where the spring is unstretched is approximately 0.326 meters.