Question

A 1.7 kg box moves back and forth on a horizontal frictionless surface between two different springs, as shown in the accompanying figure. The box is initially pressed against the stronger spring, compressing it 4.7 cm , and then is released from rest. (Figure 1)

A) By how much will the box compress the weaker spring?
B) What is the maximum speed the box will reach?

Answer 1

The box will compress the weaker spring by 4.7 cm. The maximum speed the box will reach can be calculated using the conservation of mechanical energy principle.

Step-by-step explanation:

To answer part A, we can use the principle of conservation of mechanical energy. The box starts with some potential energy stored in the stronger spring when it is pressed against it. This energy is transferred to the weaker spring when the box is released. Since the box is moving on a frictionless surface, no energy is lost to friction. Therefore, the amount of compression in the weaker spring will be equal to the initial compression in the stronger spring. So, the box will compress the weaker spring by 4.7 cm.

To answer part B, we can use the principle of conservation of mechanical energy again. The maximum compression in the springs will correspond to the maximum potential energy stored in them. At this point, all of the potential energy will be converted to kinetic energy, resulting in the maximum speed. We can calculate the maximum speed using the equation:

1/2mv^2 = kx^2

Where m is the mass of the box, v is the velocity, k is the spring constant, and x is the maximum compression. Plugging in the given values, we get:

1/2 * 1.7 * v^2 = k * (0.047)^2

Simplifying the equation, we find:

v^2 = (k * (0.047)^2) / (1.7 * 0.5)

Taking the square root of both sides, we get:

v = sqrt[(k * (0.047)^2) / (1.7 * 0.5)]

Now, substitute the given values of k, x, and m to find the maximum speed.

Answer 2

To find the answers to the given questions, we can use the concept of conservation of mechanical energy. By equating the potential energy stored in the stronger spring to the kinetic energy gained by the box, we can determine the maximum speed of the box. Additionally, by equating the potential energy gained from the weaker spring to the potential energy stored in the stronger spring, we can calculate the compression of the weaker spring.

Step-by-step explanation:

To answer this question, we need to use the concept of conservation of mechanical energy. When the box is compressed against the stronger spring, it gains potential energy which is then converted into kinetic energy as the box moves back and forth. Since the surface is frictionless, the total mechanical energy of the system remains constant. Therefore, the potential energy stored in the stronger spring is equal to the kinetic energy gained by the box. By equating these two energies, we can solve for the maximum speed of the box.

A)

Let's start by finding the potential energy stored in the stronger spring. The formula for potential energy is PE = (1/2)kx^2, where k is the spring constant and x is the compression of the spring. Given that the stronger spring is compressed by 4.7 cm, we can calculate the potential energy stored in it.

PE_stronger = (1/2)(4.5 x 10^3 N/m)(0.047 m)^2 = 5.78 J

Since the total mechanical energy is conserved, this potential energy will be converted into kinetic energy at the maximum speed. Therefore, the kinetic energy of the box is also 5.78 J.

KE_box = 5.78 J

Using the formula for kinetic energy KE = (1/2)mv^2, where m is the mass of the box and v is its velocity, we can solve for the maximum speed.

5.78 J = (1/2)(1.7 kg)v^2

v^2 = (2 * 5.78 J) / 1.7 kg

v = sqrt((2 * 5.78 J) / 1.7 kg)

v = 2.72 m/s

Therefore, the maximum speed the box will reach is 2.72 m/s.

B)

To determine how much the box will compress the weaker spring, we can use the fact that the total mechanical energy of the system is conserved. Since the box gains potential energy from the stronger spring and converts it into kinetic energy, the same amount of potential energy needs to be gained from the weaker spring.

The potential energy stored in the weaker spring can be calculated using the formula PE = (1/2)kx^2, where k is the spring constant and x is the compression of the spring. We need to solve for x. Using the calculated potential energy and the spring constant of the weaker spring, we can find the compression.

PE_weaker = (1/2)k_weaker x^2

5.78 J = (1/2)k_weaker x^2

Since the compression of the weaker spring will be positive, we take the positive square root to find the compression.

x = sqrt((2 * 5.78 J) / k_weaker)