Physics 111: Mechanics Lecture 7

Dale Gary

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NJIT Physics Department

Potential Energy and�Energy Conservation

• Work
• Kinetic Energy
• Work-Kinetic Energy Theorem
• Gravitational Potential Energy
• Elastic Potential Energy
• Work-Energy Theorem
• Conservative and

Non-conservative Forces

• Conservation of Energy

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Definition of Work W

• The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement

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• F is the magnitude of the force
• Δ x is the magnitude of the

object’s displacement

• θ is the angle between

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Work Done by Multiple Forces

• If more than one force acts on an object, then the total work is equal to the algebraic sum of the work done by the individual forces

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• Remember work is a scalar, so

this is the algebraic sum

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Kinetic Energy and Work

• Kinetic energy associated with the motion of an object

• Scalar quantity with the same unit as work
• Work is related to kinetic energy

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Units: N-m or J

Work done by a Gravitational Force

• Gravitational Force
• Magnitude: mg
• Direction: downwards to the Earth’s center
• Work done by Gravitational Force

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Potential Energy

• Potential energy is associated with the position of the object
• Gravitational Potential Energy is the energy associated with the relative position of an object in space near the Earth’s surface
• The gravitational potential energy

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• m is the mass of an object
• g is the acceleration of gravity
• y is the vertical position of the mass relative the surface of the Earth
• SI unit: joule (J)

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Reference Levels

• A location where the gravitational potential energy is zero must be chosen for each problem
• The choice is arbitrary since the change in the potential energy is the important quantity
• Choose a convenient location for the zero reference height
• often the Earth’s surface
• may be some other point suggested by the problem
• Once the position is chosen, it must remain fixed for the entire problem

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Work and Gravitational �Potential Energy

• PE = mgy

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• Units of Potential Energy are the same as those of Work and Kinetic Energy

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Extended Work-Energy Theorem

• The work-energy theorem can be extended to include potential energy:

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• If we only have gravitational force, then

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• The sum of the kinetic energy and the gravitational potential energy remains constant at all time and hence is a conserved quantity

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Extended Work-Energy Theorem

• We denote the total mechanical energy by

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• Since

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• The total mechanical energy is conserved and remains the same at all times

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Problem-Solving Strategy

• Define the system
• Select the location of zero gravitational potential energy
• Do not change this location while solving the problem
• Identify two points the object of interest moves between
• One point should be where information is given
• The other point should be where you want to find out something

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Platform Diver

• A diver of mass m drops from a board 10.0 m above the water’s surface. Neglect air resistance.
• (a) Find is speed 5.0 m above the water surface
• (b) Find his speed as he hits the water

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Platform Diver

• (a) Find his speed 5.0 m above the water surface

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• (b) Find his speed as he hits the water

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Spring Force

• Involves the spring constant, k
• Hooke’s Law gives the force

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• F is in the opposite direction of displacement d, always back towards the equilibrium point.
• k depends on how the spring was formed, the material it is made from, thickness of the wire, etc. Unit: N/m.

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Potential Energy in a Spring

• Elastic Potential Energy:
• SI unit: Joule (J)
• related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position x
• Work done by the spring

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Extended Work-Energy Theorem

• The work-energy theorem can be extended to include potential energy:

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• If we include gravitational force and spring force, then

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Extended Work-Energy Theorem

• We denote the total mechanical energy by

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• Since

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• The total mechanical energy is conserved and remains the same at all times

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A block projected up a incline

• A 0.5-kg block rests on a horizontal, frictionless surface. The block is pressed back against a spring having a constant of k = 625 N/m, compressing the spring by 10.0 cm to point A. Then the block is released.
• (a) Find the maximum distance d the block travels up the frictionless incline if θ = 30°.
• (b) How fast is the block going when halfway to its maximum height?

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A block projected up a incline

• Point A (initial state):
• Point B (final state):

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A block projected up a incline

• Point A (initial state):
• Point B (final state):

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Types of Forces

• Conservative forces
• Work and energy associated with the force can be recovered
• Examples: Gravity, Spring Force, EM forces
• Nonconservative forces
• The forces are generally dissipative and work done against it cannot easily be recovered
• Examples: Kinetic friction, air drag forces, normal forces, tension forces, applied forces …

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Conservative Forces

• A force is conservative if the work it does on an object moving between two points is independent of the path the objects take between the points
• The work depends only upon the initial and final positions of the object
• Any conservative force can have a potential energy function associated with it
• Work done by gravity
• Work done by spring force

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Nonconservative Forces

• A force is nonconservative if the work it does on an object depends on the path taken by the object between its final and starting points.
• The work depends upon the movement path
• For a non-conservative force, potential energy can NOT be defined
• Work done by a nonconservative force

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• It is generally dissipative. The dispersal

of energy takes the form of heat or sound

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Extended Work-Energy Theorem

• The work-energy theorem can be written as:

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• W_nc represents the work done by nonconservative forces
• W_c represents the work done by conservative forces
• Any work done by conservative forces can be accounted for by changes in potential energy

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• Gravity work

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• Spring force work

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Extended Work-Energy Theorem

• Any work done by conservative forces can be accounted for by changes in potential energy

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• Mechanical energy includes kinetic and potential energy

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Problem-Solving Strategy

• Define the system to see if it includes non-conservative forces (especially friction, drag force …)
• Without non-conservative forces

• With non-conservative forces

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• Select the location of zero potential energy
• Do not change this location while solving the problem
• Identify two points the object of interest moves between
• One point should be where information is given
• The other point should be where you want to find out something

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Conservation of Mechanical Energy

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• A block of mass m = 0.40 kg slides across a horizontal frictionless counter with a speed of v = 0.50 m/s. It runs into and compresses a spring of spring constant k = 750 N/m. When the block is momentarily stopped by the spring, by what distance d is the spring compressed?

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Changes in Mechanical Energy for conservative forces

• A 3-kg crate slides down a ramp. The ramp is 1 m in length and inclined at an angle of 30° as shown. The crate starts from rest at the top. The surface friction can be negligible. Use energy methods to determine the speed of the crate at the bottom of the ramp.

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Changes in Mechanical Energy for Non-conservative forces

• A 3-kg crate slides down a ramp. The ramp is 1 m in length and inclined at an angle of 30° as shown. The crate starts from rest at the top. The surface in contact have a coefficient of kinetic friction of 0.15. Use energy methods to determine the speed of the crate at the bottom of the ramp.

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fk

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Changes in Mechanical Energy for Non-conservative forces

• A 3-kg crate slides down a ramp. The ramp is 1 m in length and inclined at an angle of 30° as shown. The crate starts from rest at the top. The surface in contact have a coefficient of kinetic friction of 0.15. How far does the crate slide on the horizontal floor if it continues to experience a friction force.

Block-Spring Collision

• A block having a mass of 0.8 kg is given an initial velocity v_A = 1.2 m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Assuming the surface to be frictionless, calculate the maximum compression of the spring after the collision.

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Block-Spring Collision

• A block having a mass of 0.8 kg is given an initial velocity v_A = 1.2 m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Suppose a constant force of kinetic friction acts between the block and the surface, with µ_k = 0.5, what is the maximum compression x_c in the spring.

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Energy Review

• Kinetic Energy
• Associated with movement of members of a system
• Potential Energy
• Determined by the configuration of the system
• Gravitational and Elastic
• Internal Energy
• Related to the temperature of the system

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Conservation of Energy

• Energy is conserved
• This means that energy cannot be created nor destroyed
• If the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by some method of energy transfer

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Ways to Transfer Energy �Into or Out of A System

• Work – transfers by applying a force and causing a displacement of the point of application of the force
• Mechanical Waves – allow a disturbance to propagate through a medium
• Heat – is driven by a temperature difference between two regions in space
• Matter Transfer – matter physically crosses the boundary of the system, carrying energy with it
• Electrical Transmission – transfer is by electric current
• Electromagnetic Radiation – energy is transferred by electromagnetic waves

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Connected Blocks in Motion

• Two blocks are connected by a light string that passes over a frictionless pulley. The block of mass m₁ lies on a horizontal surface and is connected to a spring of force constant k. The system is released from rest when the spring is unstretched. If the hanging block of mass m₂ falls a distance h before coming to rest, calculate the coefficient of kinetic friction between the block of mass m₁ and the surface.

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Power

• Work does not depend on time interval
• The rate at which energy is transferred is important in the design and use of practical device
• The time rate of energy transfer is called power
• The average power is given by

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• when the method of energy transfer is work

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Instantaneous Power

• Power is the time rate of energy transfer. Power is valid for any means of energy transfer
• Other expression

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• A more general definition of instantaneous power

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Units of Power

• The SI unit of power is called the watt
• 1 watt = 1 joule / second = 1 kg ^. m² / s³
• A unit of power in the US Customary system is horsepower
• 1 hp = 550 ft ^. lb/s = 746 W
• Units of power can also be used to express units of work or energy
• 1 kWh = (1000 W)(3600 s) = 3.6 x10⁶ J

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Power Delivered by an Elevator Motor

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• A 1000-kg elevator carries a maximum load of 800 kg. A constant frictional force of 4000 N retards its motion upward. What minimum power must the motor deliver to lift the fully loaded elevator at a constant speed of 3 m/s?

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