The 10-lb block is pressed against the spring so as to compress it 2 ft when it is at A . If the

The 10-lb block is pressed against the spring so as to...

Key Concepts

Motion on a Frictionless Surface

When a surface is frictionless, as in the context of a smooth plane, no energy is lost to friction. This idealization allows for the straightforward application of energy conservation principles because all the stored energy (such as that in a compressed spring) will be fully converted into the kinetic energy of the moving object.

Conservation of Energy

The principle of conservation of energy states that in a closed system with no non-conservative forces (like friction) doing work, the total energy remains constant. In mechanics problems involving a compressed spring and a frictionless surface, the stored elastic potential energy is completely converted into kinetic energy, which may then be partly converted into gravitational potential and kinetic energy during projectile motion.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity alone. The analysis of projectile motion involves determining the object’s trajectory and the distance it travels horizontally before landing, using kinematic equations that account for the initial velocity and gravitational acceleration.

Hooke's Law

Hooke’s Law describes the behavior of springs, stating that the force exerted by a spring is directly proportional to its displacement from its natural (unstressed) length, within the elastic limit. This principle is crucial when analyzing problems where a spring is compressed or stretched, as it allows for quantifying the spring force and determining the stored elastic potential energy.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is compressed or stretched, calculated as one half the spring constant multiplied by the displacement squared. This energy becomes available to do work when the spring is released, which is then transferred, for example, to kinetic energy of an object.

Transcript

00:01 In this problem, we have a block pushing down against the spring, which will then launch this block up the ramp and then land somewhere at a certain distance d away from the wall.

00:13 And that's what we're trying to find in this problem.

00:17 So we're given a few pieces of information.

00:19 So we have the weight of this block, which is 10 pounds, and we can convert that to mass by dividing by 32 .2 feet per second squared.

00:30 So we get for the mass, 0 .31 kilograms.

00:35 We're also given the stiffness of the spring, which is 100 pounds per feet.

00:44 We know the distance the spring is compressed.

00:48 Let's call that sp, and that's 2 feet.

00:51 And we also know the distance that the block travels up the ramp.

00:55 Let's call that sb, and that's 4 feet.

01:00 So to find the distance, let's call that x.

01:03 So we know to find x, we need the velocity and the x direction times time.

01:09 So these are the two variables we need to find.

01:13 So because we're looking for a velocity and we know that this block will be in motion, we know that there's kinetic energy involved.

01:20 And we know that the force on the spring produces a work on the block and the weight of the block itself produces a work as well.

01:30 So because there's work involved and kinetic energy due to the motion, we can use the work energy theorem and this is to solve for the velocity we'll call v2 as it reaches up the ramp.

01:47 So we have the initial kinetic energy plus the work being done by the forces equals the final kinetic energy.

01:59 When we know that initially the block is at rest, so the initial kinetic energy is zero because the velocity is zero.

02:06 So then this becomes the work done by the forces is equal to final kinetic energy which is one half m v2 squared it's called that equation one okay so now let's look at the work being done by the forces so the only work or the only forces involved here is the force due to the spring and the force due to the weight well the force on the spring pushes the block up the ramp so we can call that positive work and so that's one half k the stiffness of the spring times the distance it's compressed s p squared now the weight of the block is pushing down against the force of the spring so that will be a negative work so we have the weight of that block times the distance it travels which we know is sp so we plug in our numbers we have 1 1 half times k which is 100 times s which is 2 square minus the weight we know 10 times a distance which is 3 so for work we get 170 joules and we can plug this back in to our equation 1 so this becomes 170 is equal to 1 half times the mass which is 0 .31 times the velocity squared and this is what we're solving for so let's solve for that v2 so we have v2 is equal to so if you divide 0 .31 by half and then that whole number you divide it to the other side you get 340 divided by point 31 take the square root to get rid of the square so for the velocity we have v2 is equal to 33 .11 meters per second.

04:44 Alright, now we have the velocity...

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