A spring (k=200 N / m) is fixed at the top of a...

A spring $(k=200 \mathrm{~N} / \mathrm{m})$ is fixed at the top of a frictionless plane inclined at angle $\theta=40^{\circ}$ (Fig. $\left.8-59\right) .$ A $1.0 \mathrm{~kg}$ block is projected up the plane, from an initial position that is distance $d=0.60 \mathrm{~m}$ from the end of the relaxed spring, with an initial kinetic energy of $16 \mathrm{~J}$. (a) What is the kinetic energy of the block at the instant it has compressed the spring $0.20 \mathrm{~m} ?$ (b) With what kinetic energy must the block be projected up the plane if it is to stop momentarily when it has compressed the spring by $0.40 \mathrm{~m}$ ?

Key Concepts

Mechanical Energy Conservation

In a frictionless system, the total mechanical energy remains constant, meaning that the energy can shift between kinetic energy, gravitational potential energy, and elastic (spring) potential energy without loss. This principle is essential for setting up energy balance equations to analyze the motion of objects in systems where multiple forms of energy interact.

Spring Potential Energy

Spring potential energy, often described by Hooke's Law, is the energy stored in a spring when it is compressed or stretched and is given by the formula ½kx², with k being the spring constant and x being the displacement from the equilibrium position. This concept is crucial when considering how much energy is stored or released by a spring during its compression or extension.

Gravitational Potential Energy on an Inclined Plane

When an object moves along an inclined plane, the change in its gravitational potential energy depends on the vertical component of the displacement. This energy change is quantified as mgh, where h is the vertical height, which is determined by the product of the distance traveled along the incline and the sine of the incline angle. Understanding this concept helps in accounting for energy changes due to elevation changes.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion, expressed as ½mv², where m is mass and v is velocity. It plays a primary role in energy conservation analyses, as changes in kinetic energy are directly equated to potential energy changes, enabling calculations of velocities and energy transfers in dynamic systems.

Transcript

00:01 For this problem on the topic of conservation of energy, at the top of a frictionless plane that is inclined at 40 degrees, we have a spring with a spring constant of 200 newtons per meter.

00:11 A 1 kilogram block is sent up the plane from an initial position that is a distance d of 0 .6 meters from the end of the relaxed spring.

00:21 The block has an initial kinetic energy of 16 joules.

00:24 We want to find the kinetic energy of the block after it has compressed the spring by 0.

00:30 0 .2 meters and we want to know what kinetic energy the block must have if it is projected up the plane and stops when it has compressed the spring by 0 .4 meters.

00:43 Now as the block is projected up the inclined plane its kinetic energy is converted to gravitational potential energy as well as elastic potential energy of the spring.

00:51 The block compresses the spring stops momentarily and then slides back down.

00:56 We let a be the starting point and the reference point for computing gravitational potential energy, that is ua is equal to zero, the first block, the block first comes into contact with the spring at b, the spring is compressed and by an additional amount, additional amount x at c, as shown in the figure.

01:14 So by energy conservation, we have ka plus ua is equal to kb plus ub, this must equal to kc plus uc and note that the potential energy u is that due to gravitational potential as well as elastic potential u g plus u s which is m g y plus a half k x squared for the spring now at the instant when xc is 0 .2 meters the vertical height yc is is equal to d plus xc times sine theta.

02:15 This is 0 .6 meters plus 0 .2 meters times the sign of 40 degrees.

02:29 This gives us 0 .514 meters.

02:33 And so we can apply the conservation of energy and k a plus the potential energy energy at a must equal to the kinetic energy at c plus the potential energy at c...

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