Further Mechanics 1-5

Question 1

1. Explain the circular motion of an object
2. What is the formula for linear speed?

Answer 1

1. An object in uniform circular motion has a constant linear speed. However, it is continuously changing direction , so it is constantly changing velocity, and therefore accelerating. The acceleration is callled the centripetal acceleration, and is perpendicular to the direction of the linear speed (it acts towards the centre of the circular path.) The centripetal force and acceleration act in the same direction.
2. Linear speed involving frequency = radius x angular speed.

Question 2

1. What is angular displacement?
2. Define radian
3. How do you convert from degrees to radians?

Answer 2

1. Angular displacement of a body in circular motion is defined as: the change in angle, in radians, of body as it rotates around a circle.
   It is the ratio of = distance travelled around the circle / radius of the circle.
2. Radian: The angle subtended at the centre of a circle by an arc equal.
3. Theta radians x (180 / pi) = theta degrees

Question 3

1. Define angular speed
2. What is the formula for angular speed?
3. What is the formula for angular speed involving frequency?
4. What can you infer from this equation?

Answer 3

1. Angular speed: The rate of change in angular displacement with respect to time.
2. w = delta theta / delta t
   Where delta theta = change in angular displacement (radians), and delta t = time interval (s).
3. w = v / r = 2 pi f = 2 pi / T
   Where: v = linear speed (m s-1), r = radius of orbit (m), T = the time period (s), f = frequency (Hz)
4. This equation shows that:
   The greater the rotation angle θ in a given amount of time, the greater the angular velocity ⍵. An object rotating further from the centre of the circle (larger r) moves with a smaller angular velocity (smaller ⍵).

Question 4

1. Define centripetal acceleration
2. What is the formula for centripetal acceleration?
3. What can be shown by combining the centripetal acceleration and linear speed formulae?

Answer 4

1. Centripetal acceleration: The acceleration of an object towards the centre of a circle when an object is in motion (rotating) around a circle at a constant speed.
2. a = v^2 / r
   Where: a = centripetal acceleration (m s^–2)
   v = linear speed (m s^–1)
   r = radius of the circular orbit (m)
3. These two equations can be combined to show that that centripetal acceleration is equal to the radius times the square of the angular speed. They can also show how the centripetal acceleration relates to the linear speed and the angular speed.

Question 5

1. Define centripetal force
2. What is the formula for centripetal force?
3. What is the relationship between centripetal force and acceleration?

Answer 5

1. The resultant force towards the centre of the circle required to keep a body in uniform circular motion. It is always directed towards the centre of the body’s rotation.
2. F = (mv^2) / r = mrw^2 = mvw
   Where:
   F = centripetal force (N)
   v = linear velocity (m s-1)
   ⍵ = angular speed (rad s-1)
   r = radius of the orbit (m)
3. Centripetal force and centripetal acceleration act in the same direction, due to Newton’s Second Law.

Question 6

Explain some examples of centripetal force

Answer 6

When a car is travelling around a roundabout, the centripetal force is friction between car tyres and the road. When a ball is attached to rope moving in a circle, the centripetal force is the tension in the rope. When the earth is orbiting the Sun, the centripetal force is provided by the gravitational force.

When an object travels in circular Morison, there is no work done, as there is no change in kinetic energy.