A 15.0-kg block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of 5.00 m/s in 0.460 s. In the process, the spring is stretched by 0.250 m. The block is then pulled at a constant speed of 5.00 m/s, during which time the spring is stretched by only 0.0500 m.

(a) Find the spring constant of the spring.
Answer in N/m

(b) Find the coefficient of kinetic friction between the block and the table.

(a) 1.00 × 103 N/m

(b) 0.340

Let's find the answers to both parts of the question step by step.

(a) Find the spring constant of the spring.

To find the spring constant, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

The formula for Hooke's law is given as:

F = -kx

where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.

In this case, the displacement (x) is given as 0.250 m, and the mass of the block (m) is given as 15.0 kg.

We can find the force (F) exerted by the spring using Newton's second law of motion:

F = ma

where a is the acceleration of the block.

The acceleration (a) can be calculated using the equation:

a = Δv / t

where Δv is the change in velocity and t is the time taken.

In this case, the change in velocity (Δv) is given as 5.00 m/s, and the time taken (t) is given as 0.460 s.

Substituting these values, we have:

a = 5.00 m/s / 0.460 s

Simplifying, we get:

a = 10.8696 m/s^2

Now, we can find the force (F) exerted by the spring using Newton's second law:

F = ma = (15.0 kg) * (10.8696 m/s^2)

Simplifying, we have:

F = 163.044 N

Since the force exerted by the spring is equal to -kx, we can equate the force to find the spring constant:

-kx = F

Substituting the values, we get:

-k * 0.250 m = 163.044 N

Simplifying, we have:

k = -163.044 N / 0.250 m

k = -652.176 N/m

However, since the spring constant should be positive, we take the absolute value of the calculated spring constant:

k = 652.176 N/m

Therefore, the spring constant of the spring is 652.176 N/m.

(b) Find the coefficient of kinetic friction between the block and the table.

To find the coefficient of kinetic friction (μk), we can use the equation:

Ffriction = μk * Fnormal

where Ffriction is the force of friction, Fnormal is the normal force, and μk is the coefficient of kinetic friction.

The normal force (Fnormal) can be calculated using the equation:

Fnormal = mg

where m is the mass of the block (15.0 kg) and g is the acceleration due to gravity (9.8 m/s^2).

Substituting the values, we have:

Fnormal = (15.0 kg) * (9.8 m/s^2)

Fnormal = 147 N

Since the block is pulled at a constant speed, the force of friction is equal to the force exerted by the spring.

Ffriction = F = 163.044 N

Substituting the values into the equation for friction, we get:

163.044 N = μk * 147 N

Simplifying, we have:

μk = 163.044 N / 147 N

μk = 1.109

Therefore, the coefficient of kinetic friction between the block and the table is approximately 1.109.

To solve this problem, we'll need to use Newton's second law and Hooke's law.

(a) To find the spring constant of the spring, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.

First, let's find the acceleration of the block. The block reaches a speed of 5.00 m/s in a time of 0.460 s. The average acceleration can be calculated using the formula:

acceleration = change in velocity / time

acceleration = (5.00 m/s - 0 m/s) / 0.460 s
acceleration = 10.87 m/s²

Now, let's calculate the net force acting on the block. The net force is the sum of the force provided by the spring and the force of friction. At constant speed, the force provided by the spring and the force of friction are equal in magnitude and opposite in direction. Therefore, we can set up the equation:

Net force = spring force - friction force

Since the block moves at constant speed, the net force is zero. Therefore:

0 = spring force - friction force

Next, let's calculate the force provided by the spring using Hooke's law:

spring force = spring constant * displacement

We are given that the spring is stretched by 0.250 m. Substituting this into the equation:

spring force = spring constant * 0.250 m

Now, let's calculate the force of friction. The force of friction can be determined by using the equation:

friction force = coefficient of kinetic friction * normal force

The normal force is the force exerted by the table on the block and is equal to the gravitational force acting on the block. The gravitational force can be calculated using:

force of gravity = mass * acceleration due to gravity

The acceleration due to gravity is approximately 9.8 m/s². Substituting the given values:

force of gravity = 15.0 kg * 9.8 m/s²

Now, let's substitute these values into the equation:

friction force = coefficient of kinetic friction * 15.0 kg * 9.8 m/s²

Since the spring force and friction force are equal:

spring constant * 0.250 m = coefficient of kinetic friction * 15.0 kg * 9.8 m/s²

Finally, we can solve for the spring constant:

spring constant = (coefficient of kinetic friction * 15.0 kg * 9.8 m/s²) / 0.250 m

Now, substitute the given values into the equation to find the spring constant.

(b) To find the coefficient of kinetic friction between the block and the table, we can use the fact that the net force is zero when the block is moving at a constant speed. The force of friction can be determined by using the equation:

friction force = coefficient of kinetic friction * normal force

We have already calculated the normal force as the force of gravity acting on the block. Given that the block has a constant speed of 5.00 m/s while the spring is stretched by 0.0500 m, we can set up the equation:

0 = spring force - friction force

Substituting in the values we know:

0 = spring constant * 0.0500 m - coefficient of kinetic friction * 15.0 kg * 9.8 m/s²

Now, solve for the coefficient of kinetic friction:

coefficient of kinetic friction = (spring constant * 0.0500 m) / (15.0 kg * 9.8 m/s²)

Substitute the calculated value of the spring constant into the equation to find the coefficient of kinetic friction.