Kinematics of Particles and Bodies

Question 1

What is the acceleration of a particle with respect to its tangential unit vector and its normal unit vector?

Answer 1

v = v u(t) where u is the unit vector
a = dv/dt * u(t) + v * du(t) / dt
du(t) / dt = d theta / dt * u(n)
a = dv/dt u(t) + v d theta / dt * u(n)

u(t) is the motion tangentially u(n) is the motion perpendicular

Question 2

What is the equation for a(small t) and what does it represent?

Answer 2

It is equal to dv / dt and it represnets the variation of the magnitude of velocity

Question 3

What is a (small n) and what does it represent?

Answer 3

It is equal to v^2 / radius and it represents the variation of direction of velocity

Question 4

What is the tangential unit vector of curvilinear motion using Cartesian Co-ordinates

Answer 4

cos(x) i + sin(x) j where x is the angle between the x axis and the direction of tangential motion

Question 5

What is the normal unit vector of curvilinear motion using Cartesian Co-ordinates?

Answer 5

-sin(x) i + cos(x) j where x is the angle between the x axis and the direction of tangential motion

Question 6

What is the length of an arc (s) on a circle with a radius R and an angle x?

Answer 6

s = Rx

Question 7

What is the angular velocity of a particle, and how do we derive it?

Answer 7

v = ds / dt = R * d theta / dt = R * Omega

Question 8

What is the angular acceleration relative to the angular velocity?

Answer 8

a = R * Omega ^ 2

Question 9

What are the two elementary motions of a rigid body?

Answer 9

• Translation
• Rotation

Question 10

What is translation?

Answer 10

It’s motion when a line attached to the rigid body stays parallel to itself during the motion

Question 11

What is the velocity and acceleration of all points in a rigid body under translation?

Answer 11

Velocity and acceleration is constant for all particles under translation

Question 12

What is the velocity and acceleration of all points in a rigid body under translation?

Answer 12

Velocity and acceleration is constant for all particles under translation

Question 13

What is the name of motion when the paths are curves other than straight lines?

Answer 13

Curvilinear motion

Question 14

What is the name of motion when the paths of all points in the body are circles?

Answer 14

Circular motion

Question 15

What is Rotation?

Answer 15

The motion when two points of a rigid body remain fixed during the motion. If two points remain fixed during the motion, then all points on the line connecting them are also fixed

Question 16

What is the equation for angular velocity?

Answer 16

It is the differentiation of the angle travelled with respect to time
d theta / dt

Question 17

If points A and B are both points within a rigid body, what is the relative velocity of B when compared to A?

Answer 17

We know that from the origin, r(B) = r(A) + r(B/A)
If we differentiate these, we can then find the velocities of these points
v(B) = v(A) + v(B/A)
Using the equations of circular motion, we can then calculate v(B/A) using the circular velocity
v(B/A) = dr(B/A) / dt = Omega X r(B/A)
Now we can do a final substitution to find v(B)

v(B) = v(A) + Omega X r(B/A)
The first part of the equation represents the translation, and the second part of the equation represents the rotation

Question 18

What is the Instantaneous Centre of Rotation?

Answer 18

The point which, at an instant of time has zero velocity

Question 19

How can you locate the Instantaneous Centre of Rotation geometrically?

Answer 19

Draw lines perpendicular from the velocities vectors of two points in the solid body, and see where they intersect

Question 20

How do you locate the Instantaneous Centre of Rotation if point B is on the perpendicular to the velocity of point A?

Answer 20

We can still use a geometric solution, but we need to know the magnitudes of the velocities of the two points

Question 21

Where is the Instantaneous Centre of Rotation if velocities A and B are equal?

Answer 21

It is located at infinity. This also means that the motion is a translation