A 1.40 -kg block slides with a speed of 0.950 m / s on a frictionless horizontal surface until i

A 1.40 -kg block slides with a speed of 0.950 m / s on a...

A 1.40 -kg block slides with a speed of 0.950 $\mathrm{m} / \mathrm{s}$ on a frictionless horizontal surface until it encounters a spring with a force constant of 734 $\mathrm{N} / \mathrm{m} .$ The block comes to rest after compressing the spring 4.15 $\mathrm{cm} .$ Find the spring potential energy, $U,$ the kinetic energy of the block, $K,$ and the total mechanical energy of the system, $E,$ for compressions of (a) $0 \mathrm{cm},($ b) $1.00 \mathrm{cm},(\mathrm{c}) 2.00 \mathrm{cm},$ (d) $3.00 \mathrm{cm},$ and $(\mathrm{e}) 4.00 \mathrm{cm} .$

Key Concepts

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (such as friction), the total mechanical energy of a system remains constant. In a system where kinetic and potential energies are present, energy will simply transform from one form to another, but the sum of these energies does not change over time.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object and is given by the formula (1/2)mv², where m is the mass and v is the velocity of the object. It represents the energy that an object possesses due to its motion.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is deformed by compression or extension. This stored energy can be calculated using the formula (1/2)kx², where k is the spring constant and x is the displacement from the spring's equilibrium position.

Energy Transformation

Energy transformation refers to the process by which energy changes from one form to another. In mechanical systems involving springs, this typically involves the conversion of kinetic energy into potential energy and vice versa, illustrating how energy is conserved while transitioning between different forms.

Transcript

00:01 We're told that a 1 .4 kilogram block slides without friction along a surface at 0 .95 meters per second, and then hits a spring, and come in contact with the spring at, i'm assuming this has no mass, at zero, that has a spring constant of 734 new meters per meter.

00:28 They tell us when the spring displacement is this is this is 4 .15 centimeters that the velocity is zero.

00:38 In fact, we don't actually ever use that.

00:41 We could actually use it to find k.

00:44 And in fact, there's redundant information in here.

00:50 And fortunately, it is consistent.

00:55 So if this is what i have some mass here, then we'd have to do a conservation.

01:01 Of linear momentum problem first, before we figured out, to figure out how fast the block was going after the collision.

01:08 But because we're assuming this has no mass, then the block is going the same speed just after the collision.

01:15 So just after the collision, the displacement of the spring is zero, which means the potential energy in the spring is zero, which means that all of the energy is stored kinetic energy in the block here, and that is 0 .632 joules.

01:32 So the total energy is also the kinetic energy.

01:36 Now, they ask us for about different displacements.

01:40 So as we go to one centimeter, the potential energy now is 0 .0367 joules.

01:50 And the kinetic energy is just the total energy that is stored in the system, which we have here, minus the new potential energy that is 0 .595 jules.

02:02 So again, this is a conservative system.

02:05 So the total energy has to stay the same...

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