Kinetic energy

Energy Energy is usually defined as the capacity to do work. One type of energy is kinetic energy.

Energy Energy is usually defined as the capacity to do work. One type of energy is kinetic energy. The kinetic energy is that energy associated with the motion of an object.

Energy Energy is usually defined as the capacity to do work. One type of energy is kinetic energy. The kinetic energy is that energy associated with the motion of an object. Equation: The faster an object moves, the larger its kinetic energy

How do we get an object to increase its velocity, therefore increasing its KE?

An external force must be applied to the body. Ie. Work must be done.

How do we get an object to increase its velocity, therefore increasing its KE? An external force must be applied to the body. Ie. Work must be done. Consider, v0v0 vfvf FF d A constant force F on a box accelerates it so that its velocity increases. The work done is give by the force x displacement. The work done increases the Kinetic energy of the box.

For the general case we can consider a constant force acting at an angle θ. v0v0 vfvf F d

v0v0 vfvf F d (Work done by a constant force)

The work energy theorem describes the transfer of energy. The energy associated with the work done by a force is not lost, it is transferred to kinetic energy. So that, Work-kinetic energy theorem

Work done by the Gravitational force

Consider an object moving upward through the Earth’s gravitational field. Work is done! The work is a result of a change in height.

Consider, v v0v0 d FgFg KiKi KfKf A block of mass m is thrown upward through a displacement d. The gravitational force F g acts downward on the block. The kinetic energy changes through the motion of the block. FgFg

The work done by the gravitational force F g is

As the block goes upward the displacement vector and F g act in opposing directions.

The work done by the gravitational force F g is As the block goes upward the displacement vector and F g act in opposing directions. As block falls the displacement vector and F g act in the same direction. As an object falls work is done on the block by gravity. Ie energy is transferred to the object.

As the block rises and falls, there is a change in kinetic energy and the gravitational potential energy.

Example An object of mass m = 2.0 kg is released from rest at the top of a frictionless incline of height 3 m and length 5 m. Taking g = 10 m/s 2, use energy considerations to find the velocity of the object at the bottom of the incline.

Sol:

Work done by variable forces

So far we have looked at the work done by a constant applied force.

We will now look at variable applied forces. An example of a variable force is the force applied by a spring.

So far we have looked at the work done by a constant applied force. We will now look at variable applied forces. An example of a variable force is the force applied by a spring. To illustrate the work done by a variable force we look at the work done by a spring.

Consider a spring in its equilibrium position, x=0 The force on the spring F x is zero.

Consider a spring in its equilibrium position, The spring force is given by Hooke’s law, x=0 The force on the spring F x is zero.

Consider a spring in its equilibrium position, The spring force is given by Hooke’s law, x=0 The force on the spring F x is zero.

Consider a spring in its equilibrium position, The spring is now stretched, x=0 The force on the spring F x is zero. x=0x=d d FxFx The force F x acts in the direction opposite to that of the displacement. F x is called the restoring force.

A compressed spring, x=0 The force on the spring F x is zero. x=0x=-d d FxFx The force F x now acts to the right (The direction opposite to that of the displacement).

Since the spring force ( as defined by Hooke’s law) is depends on the displacement x, the force is not constant.

Again consider, FxFx xixi xfxf The block is pulled to the right stretching the spring.

Again consider, To find the work done by the spring we integrate the force over distance moved. FxFx xixi xfxf The block is pulled to the right stretching the spring.

Again consider, FxFx xixi xfxf The block is pulled to the right stretching the spring.

Again consider, FxFx xixi xfxf The block is pulled to the right stretching the spring.

Again consider, FxFx xixi xfxf The block is pulled to the right stretching the spring.

Work done by the spring,

If the block’s final position is closer to the relaxed motion than its initial position, Work is positive.

Example A block of mass m=0.4kg slides across a frictionless surface with a speed. It then runs into and compresses a spring (k=750N/m). What distance d is the spring compressed when the block is momentarily stopped?

Example A block of mass m=0.4kg slides across a frictionless surface with a speed. It then runs into and compresses a spring (k=750N/m). What distance d is the spring compressed when the block is momentarily stopped? x=0 v

The block has an initial kinetic energy K i. The kinetic energy is zero when the block is stopped. The energy transferred from KE is used to do work in compressing the spring.

The block has an initial kinetic energy K i. The kinetic energy is zero when the block is stopped. The energy transfer is used to do work in compressing the spring. Energy is conserved.

The block has an initial kinetic energy K i. The kinetic energy is zero when the block is stopped. The energy transfer is used to do work in compressing the spring. Energy is conserved.

Power

A discussion of Work is not complete without mentioning Power.

Power is defined as the rate at which work is done by a force.

A discussion of Work is not complete without mentioning Power. Power is defined as the rate at which work is done by a force. (Average Power) (Instantaneous Power)

We can write the power in terms of force. (work by a variable force over a small displacement dx)

We can write the power in terms of force. (work by a variable force over a small displacement dx)

We can write the power in terms of force. (work by a variable force over a small displacement dx)

Example A force F is used to slide a box along a frictionless floor at constant speed of 3m/s. Calculate the Power due to the force at that instant. F=4N

Sol: