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### Integrating eat

2nd Law

A 72kg person stands on a scale which sits on a floor of elevator. It starts to move from rest upward with speed v(t) = 3t + 0.2t2,

a) find scale reading at t = 4.0s.

b) If elevator is at x =10m when t =2.0s, find location at t = 4.0s

A circus performer of weight W is walking along a "high wire" as shown. The wire bends and makes a shallow angle below the horizontal. The tension in the wire is:

a) depends on whether he stands on one or two feet

b) much more than W

c) much less than W

d) approximately W/2

e) approximately W

System Acceleration

A block of mass M and a block of mass m are connected by a thin string that passes over a light frictionless pulley. Find the acceleration of the system using only variables and constants.

Three blocks are connected as shown. Find the tension in the rope connecting the block on table. Assume no friction & massless rope.

Find system acceleration using only variables and constants.

Now find tension

example

An object of mass 5.0 kg is subjected to a rightward force in newtons of F = 3t2 – 4t where t is measured in seconds. The object has velocity v = 7.0m/s, left at t = 2.0s. Determine the velocity & acceleration of the object at t = 9.0s.

A 7.5kg shopping cart is accelerated up the incline at a rate of 1.4m/s2. Incline is angled at 13o and no friction is present.

- Determine the value of the force, F, in the horizontal direction (0o) needed to do this.
- Determine value of normal force on cart

F

The diagram shows four choices for the direction of a force of magnitude F to be applied to a block on an inclined plane. The directions are either horizontal or vertical. (For choices a and b, the force is not enough to lift the block off the plane.) Rank the choices according to the magnitude of the normal force on the block from the plane, greatest first.

Friction (f)

A force that acts parallel to surfaces in contact with one another. Two types are kinetic and static.

FAPP

fs

Static friction > Kinetic friction

Example

A 10 kg box is pulled with a force of 30N along a rough floor (μk = 0.30) as shown.

20o

- Draw FBD of ALL forces acting on box
- Determine the acceleration of the box.

Determine the acceleration of the masses assuming coefficient of kinetic friction is 0.2.

In random pairs, determine and submit on paper your answer with justification.

U-Substitution

There are times when the power rule is not an option for use as an integration technique.

Example:

For times greater than 0, an object beginning at the origin moves in one dimension according to the following expression:

Find the distance traveled by the object during the first 10 seconds.

Differential Equations

An equation that relates a quantity and its derivatives is called a differential equation. In APC Physics, we only need to be familiar with a few basic Diff E q’s. The independent variable will always be time, t.

The acceleration of an object is given by the following function:

Derive an expression for the velocity as a function of time assuming initial conditions are that x = 0 and v = 4m/s at t=0.

EXAMPLE: The acceleration of a particle,

a = –1.2x–0.8x3 and when x = +5m, v = 4m/s. Determine how far does the particle travel before coming to rest? (note ‘a’ is given in terms of x)

2 EXAMPLES

A particle moves in one dimension with a velocity as follows:

v(t) = 5e-t/2

Find the displacement of the particle from t = 0 to t = 4s.

An object falls from rest from a large distance. Due to gravity and air drag, the velocity of object follows the equation:

v(t) = 60 (1-e-t/10)

Find an expression for the displacement as a function of time.

Fdrag

Take down to be positive

mg

Force proportional to vConsider a mass (m) that experiences a resistive force of air, F = -bv, where b is positive constant. Find v as function of t.

If a force F(t) = Foe-kt acts on mass, m, that is initially at rest, what is the expression for the final velocity of object as time approaches infinity?

Bosun Chair

The tension is half in the picture on the left of what it is on the right since chair is supported by 2 ropes instead of 1

Is there a difference in the tension in the rope attached to the chair in the 2 situations? If so, what?

Assume massless, frictionless pulley & massless rope.

With what force must the man pull on the rope to rise at constant speed?

Force on rope

Circular Motion(FNET=ma for circles)

Centripetal Force, FC , is a special way to say FNET

If an object is experiencing uniform circular motion, is it in equilibrium?

Consider amusement park ride of the rotating swings. Swingcars rise into air until they reach a steady height. Find the speed of each swingcar if chain length is 5.0m and the angle btw vertical and chain is 35o

The RotoRide involves spinning people in ac circle like the spin cycle of a washing machine. Once the ride gets upto speed, the floor drops and you don’t slide. Find minimum period for this to occur assuming the radius of the ride is R and coefficient of static friction is uS.

Banked Curve

Force analysis of car without friction

What force provides FC on car to be able to perform circular motion?

What speed does car need to execute curve without friction assuming radius of curve is R and angle of curve is θ?

If friction needs to act (recall friction doesn’t always need to be present), which direction will it point?

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