A block with mass m = 14 kg rests on a frictionless table and is accelerated by a spring with spring constant k = 4545 N/m after being compressed a distance x1 = 0.515 m from the spring’s unstretched length. The floor is frictionless except for a rough patch a distance d = 2.8 m long. For this rough path, the coefficient of friction is ìk = 0.4.

How much work is done by the spring as it accelerates the block?

What is the speed of the block right after it leaves the spring?

How much work is done by friction as the block crosses the rough spot?

What is the speed of the block after it passes the rough spot?

Instead, the spring is only compressed a distance x2 = 0.151 m before being released.
How far into the rough path does the block slide before coming to rest?

What distance does the spring need to be compressed so that the block will just barely make it past the rough patch when released?

1) We know that the work done by the spring on the block is given by:

W = (1/2) * k * x1^2

Let's plug in the given values:
W = (1/2) * 4545 * (0.515)^2
W = 605.39 J

So 605.39 J of work is done by the spring as it accelerates the block.

2) We can use the work-energy principle, which states:
W = (1/2) * m * v^2

We have W = 605.39 J and m = 14 kg, so we can solve for v:
605.39 = (1/2) * 14 * v^2
v^2 = 605.39 * 2 / 14
v^2 = 86.482
v = sqrt(86.482)
v = 9.299 m/s

The speed of the block right after it leaves the spring is 9.299 m/s.

3) The work done by friction as the block crosses the rough spot can be calculated using the formula:
W_fric = -μk * m * g * d

We have μk = 0.4, m = 14 kg, g = 9.81 m/s^2, and d = 2.8 m:
W_fric = -0.4 * 14 * 9.81 * 2.8
W_fric = -153.808 J

So friction does -153.808 J of work on the block as it crosses the rough spot.

4) To find the speed of the block after it passes the rough spot, we can use the work-energy principle again:
W_total = (1/2) * m * (v_final^2 - v_initial^2)

We know that W_total = 605.39 J - 153.808 J = 451.582 J, and v_initial = 9.299 m/s:
451.582 = (1/2) * 14 * (v_final^2 - 9.299^2)
v_final^2 = 451.582 * 2 / 14 + 9.299^2
v_final^2 = 64.512 + 86.482
v_final^2 = 150.994
v_final = sqrt(150.994)
v_final = 12.287 m/s

The speed of the block after it passes the rough spot is 12.287 m/s.

5) To find how far into the rough path the block slides before coming to rest, we can use conservation of energy. We can write:
(1/2) * k * x2^2 = Π * m * g * d

Plugging in the values:
(1/2) * 4545 * (0.151)^2 = 3.14 * 14 * 9.81 * d
45.76 = 431.572 * d
d = 45.76 / 431.572
d ≈ 0.106 m

The block slides 0.106 m into the rough patch before coming to rest.

6) We'll find the distance the spring needs to be compressed so that the block will just barely make it past the rough patch when released. For this, we can equate the work done by the spring to the work done against friction:

(1/2) * k * x^2 = μk * m * g * (d+2.8)

Plugging in the values:
(1/2) * 4545 * x^2 = 0.4 * 14 * 9.81 * (2.8+d)
2272.5x^2 = 43.542 * (2.8+d)
x^2 = 43.542 / 2272.5 * (2.8+d)
x^2 = 0.01916 * (2.8+d)

Now we need to solve for d:
0.01916 * (2.8 + d) = (0.515)^2
d ≈ 0

For this particular problem, the block will barely make it past the rough patch when the spring is compressed the entire 0.515 m.

To find the answers to these questions, we need to apply some physics concepts and equations. Let's break down each question and explain how to find the answers.

Question 1: How much work is done by the spring as it accelerates the block?

The work done by the spring can be found using the equation:

Work = (1/2) * k * x1^2

where k is the spring constant and x1 is the distance the spring is compressed. Plugging in the given values, the work done by the spring can be calculated.

Question 2: What is the speed of the block right after it leaves the spring?

To find the speed of the block, we can use the principle of conservation of mechanical energy. The potential energy stored in the spring is converted to kinetic energy of the block when it leaves the spring.

The equation to find the speed v of the block is:

(1/2) * m * v^2 = (1/2) * k * x1^2

Solving for v will give us the speed of the block.

Question 3: How much work is done by friction as the block crosses the rough spot?

The work done by friction can be calculated using the equation:

Work = friction force * distance

The friction force can be calculated using the equation:

friction force = coefficient of friction * normal force

The normal force can be calculated as the weight of the block:

normal force = mg

where m is the mass of the block and g is the acceleration due to gravity.

Question 4: What is the speed of the block after it passes the rough spot?

To find the speed of the block after passing the rough spot, we need to apply the principle of conservation of mechanical energy again. The work done by friction will reduce the kinetic energy of the block.

(1/2) * m * v^2 - Work_friction = (1/2) * m * (final velocity)^2

Simplifying this equation will give us the final velocity of the block.

Question 5: How far into the rough path does the block slide before coming to rest?

To find the distance the block slides before coming to rest, we need to calculate the work done by friction until the block stops.

The work done by friction can be calculated as:

Work_friction = friction force * distance

Using the equation for friction force mentioned earlier, we can solve for the distance.

Question 6: What distance does the spring need to be compressed so that the block will just barely make it past the rough patch when released?

To find the distance the spring needs to be compressed, we need to calculate the maximum possible work done by friction on the block. This occurs when the block just barely makes it past the rough patch without stopping.

Using the equation for work done by friction and assuming the block just reaches the end of the rough patch, we can solve for the distance to compress the spring.

To find the answers to these questions, we can follow these steps:

1. Calculate the work done by the spring.
2. Determine the speed of the block after leaving the spring.
3. Calculate the work done by friction.
4. Calculate the speed of the block after crossing the rough spot.
5. Find the distance the block slides before coming to rest.
6. Determine the distance needed to compress the spring for the block to barely make it past the rough patch.

Let's calculate each step one by one.

Step 1: Calculate the work done by the spring.
The work done by the spring can be calculated using the formula:
Work = 0.5 * k * (x1^2)

Substituting the given values:
Work = 0.5 * 4545 * (0.515^2) = 593.12 Joules

So, the work done by the spring is 593.12 Joules.

Step 2: Determine the speed of the block after leaving the spring.
The potential energy stored in the spring is converted into kinetic energy of the block. Thus, using the conservation of energy, we can find the speed of the block.

Kinetic energy = Potential energy
0.5 * m * v^2 = 0.5 * k * (x1^2)

Solving for v:
v = sqrt((k * (x1^2)) / m)
v = sqrt((4545 * (0.515^2)) / 14)
v ≈ 3.15 m/s

So, the speed of the block after leaving the spring is approximately 3.15 m/s.

Step 3: Calculate the work done by friction.
The work done by friction can be calculated using the formula:
Work = friction force * distance

The friction force on the block can be calculated as:
friction force = m * g * ìk
friction force = 14 * 9.81 * 0.4 = 54.89 N

Now, using the calculated friction force and the given distance:
Work = 54.89 * 2.8 = 153.65 Joules

So, the work done by friction as the block crosses the rough spot is 153.65 Joules.

Step 4: Calculate the speed of the block after passing the rough spot.
The final kinetic energy after the block crosses the rough spot can be calculated using the work done by friction:

Kinetic energy = Potential energy - Work done by friction
0.5 * m * v^2 = 0.5 * k * (x1^2) - Work done by friction

Rearranging and solving for v:
v = sqrt((0.5 * k * (x1^2) - Work done by friction) / m)
v = sqrt((0.5 * 4545 * (0.515^2) - 153.65) / 14)
v ≈ 2.03 m/s

So, the speed of the block after passing the rough spot is approximately 2.03 m/s.

Step 5: Find the distance the block slides before coming to rest.
The work done by friction will cause the block to lose all its kinetic energy and come to rest. We can use the work-energy principle to find this distance.

Work = friction force * distance
Work = (friction force * distance) + (friction force * d)
0 = (friction force * distance) + (friction force * d)

Solving for distance:
distance = -(friction force * d) / friction force
distance = -(54.89 * 2.8) / 54.89
distance ≈ -2.8 m

The negative sign indicates that the block moves backward for a distance of approximately 2.8 m before coming to rest.

Step 6: Determine the distance needed to compress the spring for the block to barely make it past the rough patch when released.
To determine this distance, we need to find the maximum distance the block can travel before it stops due to the work done by friction.

The maximum work done by friction can be calculated as:
Max Work = friction force * max distance
Max Work = 54.89 * (d + x2)

Setting the maximum work equal to the work done by the spring:
0.5 * k * (x2^2) = 54.89 * (d + x2)

Solving for d:
d = (0.5 * k * (x2^2) / 54.89) - x2
d = (0.5 * 4545 * (0.151^2) / 54.89) - 0.151
d ≈ 0.25 m

So, the spring needs to be compressed by approximately 0.25 m for the block to barely make it past the rough patch when released.