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
a. energy available 20J
20J=10d+1/2 k d^2 k is given, calculate distance d.
b. KE=20-2*10d
c. solve for d1: 20-2*10dabove-1/2 k d1^2
d. you do it just the same...
To solve this problem, we can use the principle of conservation of mechanical energy. The mechanical energy of the system remains constant if there are no external forces doing work. In this case, the only external force doing work is the frictional force.
Let's begin with part (a):
(a) To find how far the cookie will slide before coming momentarily to rest, we need to find the potential energy stored in the spring when the cookie has zero kinetic energy.
When the cookie comes to rest, all its kinetic energy is converted into potential energy stored in the spring and work done against friction.
The potential energy stored in the spring can be calculated using the formula:
Potential energy = (1/2) * k * (displacement)^2
Where k is the spring constant and displacement is the distance the cookie has compressed or stretched the spring.
Since the problem states that the cookie is sliding on a horizontal surface, there is no vertical motion involved. Therefore, the change in height is zero and the displacement is equal to the compression or stretch of the spring.
We can set up an equation to find the displacement:
Potential energy = Work done against friction + Potential energy stored in the spring
(1/2) * k * (displacement)^2 = static friction * displacement
Rearranging the equation:
(1/2) * k * (displacement)^2 - static friction * displacement = 0
Plugging in the given values:
(1/2) * 375 * (displacement)^2 - 10 * displacement = 0
Divide both sides of the equation by displacement:
(1/2) * 375 * displacement - 10 = 0
Solving for displacement:
(1/2) * 375 * displacement = 10
375 * displacement = 20
displacement = 20 / 375
displacement ≈ 0.05333 meters
Therefore, the cookie will slide approximately 0.05333 meters before coming momentarily to rest.
Next, let's move on to part (b):
(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 work done by the spring and the work done against friction.
Initially, the cookie had a kinetic energy of 20.0 J. As the cookie slides back, the spring will do work on the cookie, converting the potential energy stored in the spring into kinetic energy. The work done by the spring will be equal to the change in potential energy of the spring.
The work done by the spring can be calculated using the formula:
Work done by spring = -(Change in potential energy)
Since the displacement is the same as the previous calculation, the change in potential energy will be:
Change in potential energy = (1/2) * k * (displacement)^2 - (1/2) * k * (initial displacement)^2
Change in potential energy = (1/2) * 375 * (displacement)^2 - (1/2) * 375 * (0.05333)^2
Change in potential energy ≈ 7.086 J
The work done by the spring is negative, so the kinetic energy of the cookie as it slides back through the position where the spring is unstretched will be:
Kinetic energy = initial kinetic energy + work done by spring + work done against friction
Kinetic energy = 20.0 J + (-7.086 J) + (-10.0 J)
Kinetic energy ≈ 2.914 J
Therefore, the kinetic energy of the cookie as it slides back through the position where the spring is unstretched will be approximately 2.914 J.
To find the answer to these questions, we need to consider the energy balance and the forces acting on the cookie.
(a) To find the distance the cookie slides before coming to rest, we need to consider the conservation of energy. The initial kinetic energy of the cookie is 20.0 J, and we need to determine how much work is done by the frictional force to bring the cookie to rest.
The work done by friction can be calculated using the equation: Work = Force × distance × cos(theta), where theta is the angle between the force and the direction of motion. In this case, the frictional force is acting against the motion of the cookie, so theta is 180 degrees.
The work done by friction is given by: Work = - Frictional force × distance.
Since the work done by friction is equal to the change in kinetic energy, we can set up the following equation:
- Frictional force × distance = change in kinetic energy
Plugging in the values, we have:
- 10.0 N × distance = -20.0 J
Solving for distance, we get:
distance = (-20.0 J) / (-10.0 N) = 2.0 m
Therefore, the cookie will slide 2.0 meters from the position where the spring is unstretched before coming momentarily 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 forces acting on the cookie again.
Since the cookie is sliding in the opposite direction, the frictional force will act in the opposite direction as well. However, the magnitude of the frictional force remains the same at 10.0 N.
Again, we use the work-energy principle to calculate the kinetic energy. The work done by the frictional force will be equal to the change in kinetic energy.
The work done by friction is given by: Work = - Frictional force × distance
Since the frictional force is acting against the motion, the angle theta is 180 degrees. Plugging in the values, we have:
- 10.0 N × distance = change in kinetic energy
As the cookie comes to rest and then slides back through the position where the spring is unstretched, it goes through the same distance as before, but in the opposite direction. Therefore, the distance is still 2.0 meters.
Plugging in the values, we have:
- 10.0 N × 2.0 m = -20.0 J
Simplifying, we get:
- 20.0 J = -20.0 J
Therefore, the kinetic energy of the cookie as it slides back through the position where the spring is unstretched is -20.0 J. Note that the negative sign indicates that the motion is in the opposite direction.
Hence, the kinetic energy of the cookie as it slides back through the position where the spring is unstretched is -20.0 J.