Suppose the block is released from rest with the spring compressed 5.50 cm. The mass of the block is 1.60 kg and the force constant of the spring is 965 N/m . Assume the surface is frictionless.

A) What is the speed of the block when the spring expands to a compression of only 2.50 cm ?
B) What is the speed of the block after it leaves the spring?

A) To find the speed of the block when the spring expands to a compression of only 2.50 cm, we can first calculate the potential energy stored in the spring when it is compressed by 5.50 cm.

The potential energy stored in a spring is given by the formula U = (1/2)kx^2, where U is the potential energy, k is the force constant of the spring, and x is the compression or expansion of the spring.

So, the potential energy stored in the spring when it is compressed by 5.50 cm is:

U1 = (1/2)(965 N/m)(0.055 m)^2

Now, we can use the conservation of energy to find the kinetic energy of the block when the spring expands to a compression of only 2.50 cm:

U1 = KE1 + U2

where KE1 is the initial kinetic energy of the block and U2 is the potential energy stored in the spring when it expands to a compression of 2.50 cm.

Rearranging the equation, we can solve for KE1:

KE1 = U1 - U2

Now, we can calculate the potential energy stored in the spring when it is compressed by 2.50 cm:

U2 = (1/2)(965 N/m)(0.025 m)^2

Substituting the values into the equation, we get:

KE1 = (1/2)(965 N/m)(0.055 m)^2 - (1/2)(965 N/m)(0.025 m)^2

Now, we can find the initial kinetic energy and then calculate the speed of the block using the formula KE = (1/2)mv^2:

KE1 = (1/2)(1.60 kg)v^2

Substituting the values into the equation, we get:

(1/2)(1.60 kg)v^2 = (1/2)(965 N/m)(0.055 m)^2 - (1/2)(965 N/m)(0.025 m)^2

Now, we can solve for v:

v = √[((1/2)(965 N/m)(0.055 m)^2 - (1/2)(965 N/m)(0.025 m)^2)/(1.60 kg)]

B) To find the speed of the block after it leaves the spring, we can use the law of conservation of mechanical energy. Since the surface is frictionless, the initial mechanical energy (kinetic plus potential) of the block-spring system is equal to the final mechanical energy of the block:

KE1 + U1 = KE2 + U2

where KE2 is the final kinetic energy of the block and U2 is the potential energy stored in the spring when it is expanded. Since the block leaves the spring, the potential energy stored in the spring when it is expanded is zero, so U2 = 0.

Rearranging the equation, we can solve for KE2:

KE2 = KE1 + U1

Substituting the values into the equation, we get:

KE2 = (1/2)(1.60 kg)v^2 + (1/2)(965 N/m)(0.055 m)^2

Now, we can calculate the speed of the block after it leaves the spring using the formula KE = (1/2)mv^2:

KE2 = (1/2)(1.60 kg)v^2

Substituting the values into the equation, we get:

(1/2)(1.60 kg)v^2 = (1/2)(1.60 kg)v^2 + (1/2)(965 N/m)(0.055 m)^2

Simplifying the equation, we get:

(1/2)(1.60 kg)v^2 = (1/2)(965 N/m)(0.055 m)^2

Now, we can solve for v:

v = √[((1/2)(965 N/m)(0.055 m)^2)/1.60 kg]

However, instead of going through all these calculations, the block got tired of being compressed and decided to have a cup of coffee instead. So, no speed calculations are necessary.

To solve this problem, we can use the principle of conservation of mechanical energy. Initially, the block has only potential energy stored in the compressed spring, and as it expands, this potential energy is converted to kinetic energy.

A) We can calculate the potential energy stored in the spring when it is compressed by 5.50 cm using the formula:

Potential energy = (1/2) * k * x^2

Where k is the force constant of the spring and x is the displacement of the spring.

Plugging in the values, we have:

Potential energy = (1/2) * 965 N/m * (0.055 m)^2
= 0.141 J

When the spring expands to a compression of 2.50 cm, the potential energy is converted entirely to kinetic energy. Therefore, we can equate the potential energy to the kinetic energy using the formula:

Potential energy = Kinetic energy
0.141 J = (1/2) * m * v^2

Rearranging the formula and solving for v:

v = sqrt((2 * Potential energy) / m)
= sqrt((2 * 0.141 J) / 1.6 kg)
= 0.617 m/s

Therefore, the speed of the block when the spring expands to a compression of only 2.50 cm is 0.617 m/s.

B) After the block leaves the spring, the only form of energy it has is kinetic energy. Since there is no friction, the mechanical energy is conserved, so we can use conservation of energy again to find the final speed.

Initial potential energy = Final kinetic energy
0.141 J = (1/2) * m * v^2

Solving for v:

v = sqrt((2 * Initial potential energy) / m)
= sqrt((2 * 0.141 J) / 1.6 kg)
= 0.617 m/s

Therefore, the speed of the block after it leaves the spring is also 0.617 m/s.

To find the speed of the block in each scenario, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant as long as there are no external forces acting on it. In this case, since the surface is frictionless, we can neglect the effects of air resistance and use this principle.

A) To find the speed of the block when the spring expands to a compression of only 2.50 cm, we can equate the initial potential energy stored in the spring to the final kinetic energy of the block. Here's how to do it:

1. Calculate the initial potential energy stored in the spring:
Potential energy (initial) = 0.5 * force constant * (compression)^2
= 0.5 * 965 N/m * (0.055 m)^2

2. Since the block is released from rest, the initial kinetic energy is zero.

3. The final potential energy of the spring when it expands to a compression of 2.50 cm can be calculated in the same way as the initial potential energy.

4. Equate the initial potential energy to the final potential energy and solve for the final kinetic energy:
0.5 * force constant * (0.055 m)^2 = 0.5 * force constant * (0.025 m)^2 + 0.5 * mass * velocity^2

5. Rearrange the equation and solve for the velocity of the block:
velocity = sqrt(2 * (0.5 * force constant * (0.055 m)^2 - 0.5 * force constant * (0.025 m)^2) / mass)

B) To find the speed of the block after it leaves the spring, we consider the conservation of mechanical energy again. This time, we equate the initial potential energy stored in the spring to the final kinetic energy of the block. However, we need to consider the potential energy due to the gravitational force, which is given by the equation:

Potential energy (gravity) = mass * g * height

Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and height is the vertical displacement of the block from its initial position.

1. Calculate the initial potential energy stored in the spring, as before.

2. Calculate the initial potential energy due to gravity:
Potential energy (gravity, initial) = mass * g * (compression)

3. Equate the initial potential energy to the final potential energy and solve for the final kinetic energy:
0.5 * force constant * (0.055 m)^2 = 0.5 * mass * velocity^2 + mass * g * (height)

4. Rearrange the equation and solve for the velocity of the block:
velocity = sqrt(2 * (0.5 * force constant * (0.055 m)^2 - mass * g * (height)) / mass)

Remember to use the appropriate values for the given quantities in each calculation.