# Exercises on work and energy

Here is available the collection of exercises solved on energy and work, physical quantities central to dynamics and of great importance in all branches of modern physics.

Thanks to these we are able to describe how the world around us moves, transforms, and how its elements interact with each other.

In this collection of exercises we will study the concepts of work and energy and see how these quantities manifest themselves and interconnect, including through familiar objects from previous collections such as springs and inclined planes.

We will also explore the different forms of energy and how they transform into each other, respecting the principle of conservation of energy. From kinetic energy to potential energy and vice versa, you will get to observe how energy flows and transforms into various forms during physical interactions.

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## Exercises solved on work and energy

### On the calculation of work - Case 1

A crate is pushed for a stretch by a force of intensity , as shown in the figure

- Calculate the work done on the case

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### On the calculation of work - Case 2

A crate is pushed for a stretch by a force of intensity forming an angle With the horizontal, as shown in the figure

- Calculate the work done on the case

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### On the calculation of kinetic energy

A mass case travels with speed

- Calculate its kinetic energy

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### On the calculation of work and the live force theorem

A massive machine goes from an initial velocity to a final speed Due to the action of the brakes

- Calculate the work done by the brakes

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### On the calculation of work for a braking machine

A massive machine travels with initial speed directed as in the figure. At a certain instant it brakes for meanwhile covering a distance

- Calculate the work done by the brakes

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### On gravitational potential energy

A mass diver stands on a high trampoline with respect to the pool

- Calculate the gravitational potential energy of the diver

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### On elastic potential energy

A compressed spring of with respect to the equilibrium position stores elastic potential energy equal to

- Calculate the spring's elastic constant

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### On the conservation of mechanical energy

A mass case , initially stationary, falls from a height of

- Calculate the speed at which the crate reaches a distance

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### On mechanical energy and nonconservative forces

A mass case is launched from the base of an inclined plane of with initial velocity . There is friction between the case and the inclined plane with a dynamic friction coefficient of

- Calculate at what height crate stops

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### On mechanical energy and work

A mass cloth falls from a balcony that is at a height of from the ground. The moment it touches the ground it has a velocity

- Calculate the work done by air resistance

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### On the spring and elastic potential energy

A spring of elastic constant is compressed by a quantity Relative to the equilibrium position. A mass case is in contact with the spring, as shown in the figure. Between the case and the spring is friction with dynamic coefficient of friction . At the initial instant the case is let go:

- Calculate the value of the compression such that the case reaches the point distant

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### On potential energy and work

A ball of mass Is launched into the air from an initial height and reaches a final height

- Calculate the work done by the force of gravity

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### On the spring and gravitational potential energy

A ball of mass Is connected to a spring of spring constant oscillating in harmonic motion with pulsation . During the oscillations, the height of the ball varies by and its gravitational potential energy varies by

- Calculate the spring's elastic constant

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### On the conservation of momentum and energy - The elastic collision

A mass white billiard ball moves with initial velocity Toward another yellow billiard ball of mass . After the impact, the white billiard ball moves with final velocity

- Calculate the speed Of the yellow billiard ball after the impact

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### In the inclined plane, the work of the nonconservative forces and the mechanical energy

A mass case initially stationary lies on a plane inclined at an angle at a height . The inclined plane, along , has a dynamic friction coefficient of until the first half of the journey and in the second half, as shown in the figure. If the body reaches the base of the inclined plane with velocity

- Calculate the dynamic coefficient of friction In the second half of the

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### On the inclined plane, the pulley and the conservation of mechanical energy

Two mass crates and are connected by an ideal rope as shown in the figure. The ground case lies on an inclined plane of With dynamic friction coefficient between the case and the plane equal to . The system, initially stationary, is allowed to move freely and the mass case falls of a stretch .

- Calculate the velocity of the system formed by the two cases after the case of mass falls of

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### On the pendulum and the principle of conservation of energy

The pendulum shown in the figure is formed by an ideal rope of length attached to a mass weight and is free to swing around the hinged end and indicated in the figure by the point . At the lowest point the kinetic energy of the weight is

- Calculate the tension of the rope When the pendulum is in a horizontal position

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### On the work of the frictional force

A mass case Is thrown on a rough horizontal plane and long with initial velocity . At the end of the horizontal plane the box rises along an inclined plane of also rough. Both the horizontal plane and the inclined plane have a dynamic friction coefficient of .

- Calculate the time necessary for the case to stop

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### On the pulsation of a spring, the mechanical energy and work of a nonconservative force

A mass case rests on a horizontal rough plane with dynamic friction coefficient between the plane and the case equal to . The case is connected to a spring of elastic constant , as shown in the figure, initially compressed by with respect to the equilibrium position. If the case, initially stationary, is allowed to move freely, it reaches the equilibrium position with velocity

- Calculate the pulsation

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### A simple problem on conservation of energy for a body along a guideway

Consider the problem shown in the figure, in which a body of mass , resting on a firm, frictionless rail, is launched with speed from a point to height Relative to the ground.

- Calculate the speed at the point

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### A problem on spring and conservation of energy without calculations - 1

Consider a body of mass moving with speed constant toward a massless buffer, as shown in the figure.

- Calculate the minimum length reached by the buffer spring

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### A problem on spring and conservation of energy without calculations - 2

Consider the previous problem** **assuming that the plane on which the body of mass rests Is rough, that is, a frictional force is present. Let us assume that the body is initially distant from the buffer and moving with initial velocity , as shown in the figure.

- Calculate the minimum length reached by the buffer spring

### On the conservation of energy and the death lap

A ball of mass Initially still slips from a height along a frictionless guideway, at the end of which there is a *"lap of death"*, as shown figure. The circular part of the guide has radius

[Hint: The ball always stays inside the guide in the circular section!]

- Calculate the height minimum from which the ball must start to arrive at the point Without detaching from the guide

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### On the kinetic energy theorem (or live force theorem)

A mass case is launched at an initial speed along a horizontal plane. Friction is present between the case and the horizontal plane,

- Calculate the work done by the frictional force to stop the case

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### On the kinetic energy of a sphere in rotation

A homogeneous sphere of mass and radius Rolls without crawling on a horizontal plane. If the velocity of the center of mass is and the angular velocity is

- Calculate the kinetic energy of the sphere

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### On the principle of conservation of energy for a rigid system

A homogeneous disk of radius and mass Is initially stationary on an inclined plane at a height from the ground, as shown in Figure

- Calculates the velocity of the center of the disk when it has reached ground level

[Hint: Assume that the disk rolls without crawling, i.e., we are in the presence of pure rolling!]

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### On the spring and the death lap

A ball of mass initially stationary is thrown by a spring of spring constant along a frictionless track that has a circular section (a "lap of death") of radius as shown in the figure

- Calculate the compression of the spring required to make the ball complete an entire lap of death without coming off the track

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