A wooden block with mass 1.50 kg is placed against a compressed spring at the bottom of an incli

A wooden block with mass 1.50 kg is placed against a...

A wooden block with mass 1.50 $\mathrm{kg}$ is placed against a compressed spring at the bottom of an incline of slope $30.0^{\circ}$ (point $A$ ). When the spring is released, it projects the block up the incline. At point $B,$ a distance of 6.00 $\mathrm{mup}$ the incline from $A$ , the block is moving up the incline at 7.00 $\mathrm{m} / \mathrm{s}$ and is no longer in contact with the spring. The coefficient of kinetic friction between the block and the incline is $\mu_{\mathrm{k}}=0.50 .$ The mass of the spring is negligible. Calculate the amount of potential energy that was initially stored in the spring.

Key Concepts

Conservation of Energy

This concept states that the total energy of a closed system remains constant, though it may transform from one form to another. In problems where a spring propels an object, energy is transferred from the stored elastic potential energy in the spring to kinetic energy, gravitational potential energy, and energy lost due to friction. This is a key principle in analyzing energy transformations in mechanical systems.

Spring Potential Energy

Spring potential energy is the energy stored in a compressed or stretched spring. It is the initial energy input in situations where a spring is used to launch an object, and it is calculated based on the work done by the spring force. In this problem, the total spring energy is converted into other forms of energy as the block moves up the incline.

Work-Energy Principle

The work-energy principle connects the work done by all forces acting on an object to its change in kinetic energy. In mechanical systems, this principle can include both conservative forces (like gravity, which are stored as potential energy) and non-conservative forces (such as friction, which dissipate energy). This principle underlies the calculation of how much energy remains as kinetic energy after accounting for losses.

Friction as a Non-Conservative Force

Friction is a resistive force that converts mechanical energy into thermal energy, dissipating energy from the system. In this problem, kinetic friction acts on the block as it moves up the incline, doing work against its motion. The energy lost due to friction must be considered when applying energy conservation to determine the initial energy stored in the spring.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. When the block ascends an incline, its gravitational potential energy increases according to the vertical height gained. This increase is a key component in the energy balance that determines the initial energy provided by the spring.

Transcript

00:01 So here we have the height of point b we respect the level of base of inclined plan is 60 meter into sign of 30 degree here we will take the base of inclined plane as y equal to zero so this gives us potential energy of point a equal to zero now we can write conservation of energy here as kinetic energy at point a plus potential energy of of potential energy a block at point a plus work done by forces other than gravity is equal to kinetic energy at point b plus potential energy at point b now by putting different values here as the block starts from yes so we have initial kinetic energy at point a equal to zero plus potential energy at point a here gravitation protection energy at point a is equal to zero plus elastic protection energy at point a is equal to zero plus elastic protection energy at point a is elastic protection energy at point a plus work done by forces other than gravity here work done by friction so work done by friction is equal to minus kinetic friction times distance moved by block on incline plan let this is d equal to kinetic energy at point b.

02:53 Kinetic energy of block at point b is half times mass of block is 1 .5 kilogram and speed of block at point b is 7 meter per second plus protection energy of block at point b here mess of block is 1 .5 kilogram gravitation acceleration is 9 .8 meter per second square and height of point b is d sign of 30 degree this gives us elastic potential energy at point a minus kinetic friction that is equal to coefficient of kinetic friction which is 0 .50 into normal reaction force here normal reaction force is equal to mass of block which is 1 .5 kilogram into gravitational acceleration is 9 .8 meter per second square into cosine of 30 degree so this is normal reaction force here one component of weight acts as normal reaction force which is equal to w cosine of theta and distance moved by block on incline plane is d that is 60 meter which is equal to kinetic energy at point b is kinetic energy at point b is 36 point b is 75 june plus potential energy at point b is 1 .5 kg...

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