2. You push a 2.0 Kg block against a horizontal spring (the spring constant is k=200 N/m)
compressing it by 15 cm. Then you release the block and the spring sends it sliding across a
tabletop. The block stops 75 cm from where you released it. (a) Find the work done by the
spring force as it pushes the object. (b) Find the work done by frictional force acting on the
object. (c) Find the work done by all the forces acting on the block. (d) What is the
coefficient of kinetic friction k?
To solve this problem, we need to find the work done by the spring force, the work done by the frictional force, the work done by all the forces, and the coefficient of kinetic friction.
(a) Work done by the spring force (W_spring):
The work done by the spring force is given by the formula:
W_spring = (1/2) * k * x^2
where k is the spring constant and x is the displacement. In this case, k = 200 N/m and x = 15 cm = 0.15 m.
W_spring = (1/2) * 200 * (0.15)^2
W_spring = 2.25 Joules
Therefore, the work done by the spring force is 2.25 Joules.
(b) Work done by the frictional force (W_friction):
The work done by the frictional force can be calculated using the formula:
W_friction = -frictional force * distance
Since the block stops after sliding on the tabletop, the work done by friction will be equal in magnitude and opposite in direction to the work done by the spring force.
W_friction = -W_spring
W_friction = -2.25 Joules
Therefore, the work done by the frictional force is -2.25 Joules.
(c) Work done by all forces (W_total):
The work done by all the forces acting on the block can be calculated by summing the work done by the spring force and the work done by the frictional force:
W_total = W_spring + W_friction
W_total = 2.25 Joules + -2.25 Joules
W_total = 0 Joules
Therefore, the work done by all forces acting on the block is 0 Joules.
(d) Coefficient of kinetic friction (μk):
The coefficient of kinetic friction (μk) can be found using the equation:
μk = f_friction / f_normal
where f_friction is the frictional force and f_normal is the normal force. Since the block is sliding horizontally on a tabletop, the normal force is equal to the weight of the block.
f_normal = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, the mass of the block is 2.0 kg.
f_normal = 2.0 kg * 9.8 m/s^2
f_normal = 19.6 N
The frictional force can be calculated using the formula:
f_friction = μk * f_normal
Substituting the known values:
-2.25 Joules = μk * 19.6 N * 0.75 m
Simplifying:
μk = -2.25 Joules / (19.6 N * 0.75 m)
μk ≈ -0.1504
Since the coefficient of kinetic friction cannot be negative, the coefficient of kinetic friction is approximately 0.1504.
Therefore, the coefficient of kinetic friction (μk) is approximately 0.1504.
To find the answers to the given questions, we will break it down step-by-step.
(a) To find the work done by the spring force, we can use the formula:
Work (W) = (1/2) * k * (x^2)
where k is the spring constant and x is the displacement. In this case, the spring constant is given as 200 N/m and the displacement is 15 cm, which we need to convert to meters. So, 15 cm = 0.15 m.
Therefore, the work done by the spring force is:
W = (1/2) * 200 * (0.15^2)
(b) To find the work done by the frictional force, we need to calculate the distance over which the friction is acting. Given that the block stops 75 cm from where you released it and assuming there is no other horizontal force acting on it, this distance is also the displacement.
The work done by frictional force can be calculated using the formula:
W = force * distance
However, we need to find the force first. We know that weight (mg) is acting vertically downwards, where m is the mass of the block (2.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The normal force (N) acting upwards must balance the weight, so N = mg.
The frictional force (f) can be calculated as f = μk * N, where μk is the coefficient of kinetic friction.
Now, we can calculate the work done by the frictional force:
W = μk * N * distance
(c) To find the work done by all the forces acting on the block, we simply need to add the work done by the spring force (part a) and the work done by the frictional force (part b).
Work done by all forces = Work done by spring force + Work done by frictional force
(d) To find the coefficient of kinetic friction (μk), we can rearrange the equation from part b:
μk = (W / N) / distance
Now that we have broken down the steps, you can plug in the values and calculate the respective answers.