PHYS 1003 Lecture Notes - Lecture 2: Unit Vector

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12 Oct 2016

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** if you are asked to find a vector quantity in a problem, it always means find both magnitude and direction. A displacement made in three dimensions is represented by a vector r (arrow on top) The vector in textbooks is represented using a bold font r. We define three unit vectors i, j, and k to represent motion in x,y and z directions. A unit vector has a magnitude of 1, and has no units. Motion of an object r = -7. 0 i metres (if going on negative x axis) We describe motion of an object with three parameters: The position of an object is given by a vector coordinate. Note that to define position, the origin of the axis must be defined. Displacement is the change in position of an object. Where the origin is defined doesn"t matter. Velocity is the rate of change of displacement with respect to time.

Related Questions

(A) Show that the position of a particle on a circle of radius Rwith center at the origin (x=0,y=0) is given in unit vectors byr=cosÎ¸i+sinÎ¸j, where Î¸ is the angle of the position vector with thex axis.

B) If the particle moves with constant speed v and period T,starting on the x axis at t=0, find an expression for Î¸ in terms ofthe time and T.

C) Differentiate the position vector twice with respect to timeto find the acceleration and show that it is the centripetalacceleration, whose magnitude is given by equation (a=(v^2)/r) andwhose direction is toward the center of the circle. (Not that theunit vectors i and j are constants, independent of time.)

Good Vibes: Introduction to Oscillations

The conditions that lead to simple harmonic motion are as follows:

There must be a position of stable equilibrium.

There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by ⃗, where is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

Consider a block of mass m attached to a spring with force constant k, as shown in the figure(Figure 1) . The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from x=A, it will

(a) remain at rest

(b) move to the left until it reaches equilibrium and stop there.

(d) move to the left until it reaches x=−A and then begin to move to the right.

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.

The frequency, denoted f, is the number of cycles that are completed per unit of time: f=1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).

Part B

If the period is doubled, the frequency is

(a) unchanged

(b) doubled

Part C

An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?

Express your answer in hertz.

Part D

If the frequency is 40 Hz, what is the period T?

Express your answer in seconds.

The following questions refer to the figure (Figure 2) that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x-coordinate of the oscillating object, and the horizontal axis represents time.

Part E

Which points on the x-axis are located at a distance A from the equilibrium position?

(a) R only

(b) Q only

Part F

Suppose that the period is T. Which of the following points on the t-axis are separated by the time interval T?

(a) K and L

(b) K and M

(d) L and N

(e) M and P

Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.

Part G

What is the period T?

Express your answer in seconds.

Part H

How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?

Express your answer in seconds.

Part I

What distance d does the object cover during one period of oscillation?

Express your answer in meters.

Part J

What distance d does the object cover between the moments labeled K and N on the graph?

Express your answer in meters.

emeraldyak680

An object of mass m has these three forces acting on it (there is no normal force, "no surface"). F1 = 7 N, F2 = 7 N, and F3 = 5 N. When answering the questions below, assume the x-direction is to the right, and the y-direction is straight upwards.

a) What is the net force in component form, in terms of F1, F2, F3, and the unit vectors i and j

b) What is the magnitude of the net force, in newtons?

c) What is the angle

, in degrees, of the net force, measured from the +x-axis? Enter an angle between -180° and 180°.

d) What is the magnitude, |a| of the acceleration, in meters per square second, if the block has a mass of 8.5 kg?

a) What is the net force in component form, in terms of F1, F2, F3, and the unit vectors i and j
b) What is the magnitude of the net force, in newtons?
c) What is the angle Î¸, in degrees, of the net force, measured from the +x-axis? Enter an angle between -180Â° and 180Â°.
d) What is the magnitude, |a| of the acceleration, in meters per square second, if the block has a mass of 6.9 kg?

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PHYS 1003 Lecture Notes - Lecture 2: Unit Vector

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