Physics Final

Equillibrium

The center of mass of an object is the point on the object that moves in the same way that a point particle would move.

• Objects that move freely through space rotate about an axis that goes through their center of mass.

• An object is said to be stable if a large external force is required to tip it.

• A box is stable as long as the direction of the torque due to its weight τ_w keeps it upright.

• This occurs as long as the box’s center of mass lies above its base.

  

  • An object is said to be in static equilibrium if both its velocity and its angular velocity are zero or constant.

  • First, it must be in translational equilibrium; that is, the net force exerted on the object must be zero. 

  • Second, it must be in rotational equilibrium; that is, the net torque exerted on the object must be zero.

• Newton’s laws are valid only in inertial or nonaccelerated frames. 

• Newton’s laws would not apply in rotating frames of reference, as they are accelerated frames.

F_{1 X} R₁₌ F_{2 X} R_{2 → Equation 4 Equillibrium}

Momentum

Momentum is defined as the mass times the velocity.

The symbol for momentum is kg.m/s p=mv

• an object's momentum changes whenever its mass or velocity changes.

• P is sometimes referred to as the linear momentum to distinguish it from angular momentum, a quantity associated with a rotating object.

• Momentum is a vector quantity. The momentum vector points in the same direction as the velocity vector.

△P=P_f-P_i

Impulse

The product of a force and the time over which it acts is defined as the impulse I=F△t

Because impulse involves the product of force and time, a small force acting over a long time has the same effect as a large force acting over a short time.

• The units of impulse are the same as the units of momentum, namely, kg.m/s

• Impulse is a vector that points in the same direction as the force.

The impulse-momentum theorem states that the impulse on an object is equal to the change in its momentum.

I=Fxt=△P

Impulse Changes Momentum

• Coming to a stop is caused by a force over a specific time

• Increasing impact time greatly reduces force

• The reason for many safety features

– Car bumpers, air bags, boxing gloves, etc.

A small impact time produces very large forces

• Why people are injured in car crashes

– Landing stiff legged compared to bent, barrel things on freeways, etc.

Newton’s second law for rotational motion (τ = I Δω/Δt) states that angular velocity of a rotating object changes only if torque is applied to it.

• This equation can be rearranged in the same way as Newton’s second law of motion was, to produce τΔt = I Δω.

• The left side of this equation is the angular impulse of the rotating object and the right side can be rewritten as Δω = ωf − ωi.

The angular momentum of an object is equal to the product of a rotating object’s moment of inertia and angular velocity.

Angular Momentum

L = I Δω

• Angular momentum is measured in kg·m²/s.

Just as the linear momentum of an object changes when an impulse acts on it, the angular momentum of an object changes when an angular impulse acts on it.

• Thus, the angular impulse on the object is equal to the change in the object’s angular momentum, which is called the angular impulse-angular momentum theorem.

Angular Impulse-Angular Momentum Theorem TΔt = Lf − Li (T→ torque)

If there are no forces acting on an object, its linear momentum is constant. If there are no torques acting on an object, its angular momentum is constant.

• Because an object’s mass cannot be changed, if its momentum is constant, then its velocity is also constant.

The angular velocity of an object can change, however, even if no torques are acting on it.

• This is because the moment of inertia depends on the object’s mass and the way it is distributed about the axis of rotation.

• For example, a diver can control her angular velocity by changing her moment of inertia.

Conservation of Momentum

A system which does not gain or lose mass is said to be a closed system.

• When the net external force on a closed system is zero, the system is described as an isolated system.

The law of conservation of momentum states that the momentum of any closed, isolated system does not change.

Law of Conservation of Momentum

p_f = p_i m₁v₁+m₂v₂=m₁v₁+m₂v₂

If F_ext = 0 then p = CONST

Law of Conservation of Momentum (stick together- inellastic collision)

m₁v₁+m₂v₂=(m₁+m₂) v_{f -kinetic energy isnt conserved}

Law of Conservation of Momentum (ellastic collision)

m₁v₁+m₂v₂=m₁v_1’+m₂v_2’

• Conservation of momentum can be used to explain recoil.

m₁v₁=-m₂v₂

Like linear momentum, angular momentum can be conserved under certain conditions.

• The law of conservation of angular momentum states that if no net external torque acts on an object, then its angular momentum does not change.

IW _before = IW _after

precession Which is the term for the rotation of the upper end of a tipped-over top as it revolves around its rotational axis.

Work

we define work W as force F times the displacement s, over which the force acts:

W = Fd

unit: (Nm) or (J)

If the force is not parallel to the displacement the formula for work has the minor correction

W = Fd cos θ

If F and d are parallel, θ = 0° and cos 0° = +1.

If F and d are antiparallel, θ = 180° and cos 180° = -1.

• When a net force is exerted on an object while the object moves a certain distance, the object will be accelerated, and its velocity will increase.

• Fd =^½ mv²_{f -} ½ mv_i²

A graph of force versus displacement lets you determine the work done by a force. This graphical method can be used to solve problems in which the force is changing.

• In the equation for work the quantity 1⁄2mv2 represents a property of the system. This property, which is the ability of an object to produce a change in itself or the world around it, is called energy (E).

• The work-energy theorem states that when work is done on a system, the result is a change in the system’s energy.

w=ke=^½ mv²_{f -} ½ mv_i²

^.

If the external world does work on the system, then W is positive, and the energy of the system increases.

• If the system does work on the external world, then W is negative, and the energy of the

system decreases.

Kinetic energy, the energy associated with motion,You can use a bar graph to represent changes in energy.

The rate at which energy is transformed is power.

P =ΔE/t

• Power is measured in watts (W). One watt is 1 Joule of energy transferred in 1 second.

• When force and displacement are in the same direction, P = Fd/t. However, the ratio d/t is the

speed, so you can also use P = Fv.

Many forms of energy

Energy comes in different varieties. Energy can be changed from one variety to another.

1. Energy due to changing position is called translational kinetic energy and can be represented by the following equation:

KE_trans=1/2mv_{Translational}²

2. Kinetic energy also can be due to rotational motion.

• The kinetic energy of a spinning object is called rotational kinetic energy.

• Rotational kinetic energy can be calculated using the system’s moment of inertia (I) and its angular velocity (ω).

KE_rot = 1⁄2Iω²

Energy that is stored due to interactions between objects in a system is called potential energy.

• The stored energy due to the gravitational force between objects is called gravitational potential energy, or GPE. GPE=mgh

• The height to which an object has risen is determined by using a reference level, the position where GPE is defined to be zero.

The stored energy due to an object’s change in shape, like a drawn bowstring, is called elastic

potential energy, which is often stored in rubber balls, rubber bands, slingshots, and trampolines.