Rotational Motion and Astrophysics

Question 1

Explain the difference between angular, tangential and centripetal acceleration.

Answer 1

• Angular acceleration is the rate of change of angular velocity
• Tangential acceleration is the rate of change of tangential speed
• Centripetal acceleration is the rate of change of tangential velocity- this causes the direction of the object to change and results in circular motion.

Question 2

How would you find the instantaneous velocity of an object from a displacement-time graph such as this?

Answer 2

Take the gradient of the line.

If the graph is a curve, take the gradient of the tangent to the curve at a point.

Question 3

How would you find the instantaneous acceleration of an object from a velocity-time graph such as this?

Answer 3

Take the gradient of the line.

If the graph is a curve, take the gradient of the tangent to the curve at a point.

Question 4

How would you find the displacement of an object from a velocity-time graph such as this?

Answer 4

Calculate the area underneath the graph- remember to include negative values.

If the graph is a curve, the equation of the graph can be integrated to get an expression for displacement.

Question 5

How would you convert an angle from degrees into radians?

Answer 5

Angle in radians = (2π/360) x Angle in degrees

Question 6

What relationships would you use to convert between:

1. Linear displacement and angular dispalcement
2. Linear velocity and angular velocity
3. Linear acceleration and angular acceleration

Answer 6

For any of these relationships, you would multiply the angular quantity by the radius of circular motion to get the linear quantity:

Question 7

How is angular velocity defined in terms of angular dispalcement?

Answer 7

Angular velocity is defined as the rate of change of angular displacement:

Question 8

How is angular acceleration defined in terms of angular velocity?

Answer 8

Angular acceleration is defined as the rate of change of angular velocity.

Question 9

What is meant by Torque (sometimes called the moment of a force)?

Answer 9

Torque is the turning effect of a force on a rotating object.

Question 10

What is meant by the moment of inertia of an object?

Answer 10

The moment of inertia of an object is a measure of its resistance to angular acceleration about a given axis.

Question 11

State the law of conservation of angular momentum.

Answer 11

In the absence of external torques, the total angular momentum of object(s) before a collision is equal to the total angular momentum of the object(s) after the collsion.

Question 12

Which two quantities does the rotatinal kinetic energy of an object depend on?

Answer 12

Moment of Inertia and angular velocity

Question 13

What is the definition of the gravitational field strength at a point?

Answer 13

The force exerted per unit mass (by a gravitational field).

Question 14

What is the definition of gravitational potential at a point in space?

Answer 14

The work done in moving a unit mass from infinity to that point in space.

Question 15

At which point do both gravitational potential and gravitational potential energy have values of zero?

Answer 15

They both have values of zero at infinity

Question 16

What is the definition of escape velocity?

Answer 16

The minimum velocity required to allow a mass to escape a gravitational field (and have zero gravitational potential energy)

or

The minimum velocity required to allow a mass to reach infinity

Question 17

By referring to ‘frames of reference’, in which situations would

a) Special relativity

and

b) General relativity

apply?

Answer 17

a) Special relativity deals with motion in inertial (non-accelerating) frames of reference.
b) General relativity deals with motion in non-interial (acceelrating) frames of reference.

Question 18

State what is meant by the ‘equivalence principle’.

Answer 18

It is impossible to tell the difference
between the effects of a uniform gravitational field and
of a constant acceleration.

Question 19

Describe the effect of placing a clock at a higher altitude in a gravitational field.

Answer 19

The clock would run faster higher in a gravitational field.

Question 20

On a spacetime diagram, what is the name given to lines representing an object’s motion?

Answer 20

World-lines

Question 21

Describe the effect of placing a clock at a lower altitude in a gravitational field.

Answer 21

The clock would run slower lower in a gravitational field.

Question 22

The world-lines on the following space time diagram represent the motion of different objects, A, B and C.

Describe the motion of objects A, B and C.

Answer 22

Object A is stationary

Object B is moving with a constant velocity

Object C is accelerating

Question 23

Light or freely moving objects under the influence of gravity follow particular paths in spacetime. What are these paths called?

Answer 23

Geodesics

Question 24

Describe the effect a mass has on spacetime.

Answer 24

Mass curves (or distorts) spacetime

Question 25

Describe what we understand as gravity in terms of spacetime.

Answer 25

Gravity is caused by the curvature of spacetime. Masses follow this curvature of spacetime.

Question 26

Stars can be split into what are known as 'spectral classes'. What are the seven spectral classes?

Answer 26

**O B A F G K M**

Question 27

Which of the seven spectral classes of stars is the hottest?

Answer 27

Spectral class **O** is the hottest.

Question 28

Which two quantities does the **Luminosity** of a star depend on?

Answer 28

The luminosity of a star depends on it's **radius** and **surface temperature**

Question 29

When stars fuse hydrogen into helium via the proton-proton chain releasing energy (gamma rays) in the process, which two **other** particles are produced?

Answer 29

positrons and neutrinos

Question 30

In this simplified Hertzsprung-Russel diagram, which types of stars can be found in areas A, B, C and D?

Answer 30

A- Main sequence stars B- (Red) Giants C- (Red) Supergiants D- White dwarfs

Question 31

Explain the stages of a low to medium mass star's evolution from when it leaves the main sequence until its eventual fate.

Answer 31

* Low to medium mass stars will stop fusing H to He, their cores will contract due to gravity and will start fusing He into heavier elements such as C and O. * They will then move into the giants region of the HR diagram. * When He fusion stops, the outer layers drift off to become a **planetary nebula**. * This leaves behind a hot, dense core known as a **white dwarf.**

Question 32

Explain the stages of high mass star's evolution from when it leaves the main sequence until its eventual fate.

Answer 32

* Higher mass stars can fuse elements in stages all the way up to Iron. * These stars produce so much thermal pressure that they move into the **supergiant** region of the HR diagram. * When fusion stops, the star's core collapses suddenly, and rebounds outwards as a violent **supernova** explosion. * The remaining core is so dense that it becomes a **neutron star** or **black hole.**

Question 33

Which quantity of a star determines its ultimate fate?

Answer 33

Its mass.

Question 34

While in the main sequence for the majority of it's lifetime, explain how a star remains stable.

Answer 34

The thermal pressure outwards produced by fusion is balanced by gravitational attraction. It is in **gravitational equilibrium.**

Question 35

An expression for the displacement of an object with respect to time is given as s = 3.1t² + 4.1t + 6 Determine exprssions for the object's velocity and acceleration.

Answer 35

v = 6.2t + 4.1 a = 6.2

Question 36

A ball of weight W attached to a string is 'whirling' in a horizontal circle as shown below. What is the centripetal force acting on the ball?

Answer 36

The centripetal force is suppled by the tenson, T only. T is the only force acting into the centre.

Question 37

A ball of weight W attached to a string under tension T is 'whirling' in a vertical circle as shown below. Determine a relationship for the centripetal force acting on the ball: 1. at the bottom. 2. at the sides. 3. at the top.

Answer 37

1. Centripetal force = Tension - Weight 2. Centripetal force = Tension 3. Centripetal force = Tension + Weight

Question 38

A 'conical pedulum' is shown below. In terms of the components of tension in the string determine expressions for the: 1. Centripetal force 2. Weight of the 'bob'.

Answer 38

1. Centripetal force = T sinθ 2. Weight = T cosθ

Question 39

In a 'loop the loop' ride at a funfair, how would you determine the minimum speed required to complete a full loop?

Answer 39

The minimum speed required is when there is no reaction force acting on the 'car', so. Centripetal force = Weight + Reaction force simplifies to Centripetal force = Weight mv²/r = mg **v = √gr**

Question 40

What is the unit of centripetal acceleration?

Answer 40

ms^-2

Question 41

What is the unit of angular acceleration?

Answer 41

rad s^-2

Question 42

What is the unit of tangential acceleration?

Answer 42

ms^-2

Question 43

What is the unit of anguar displacement?

Answer 43

Radians (rad)

Question 44

What is the unit of angular velocity?

Answer 44

rad s^-1

Question 45

What is the unit of moment of inertia?

Answer 45

kg m²

Question 46

What is the unit of Torque?

Answer 46

Newton-metres (Nm)

Question 47

What is the unit of angular momentum?

Answer 47

kilogram metres squared per second (kg m² s^-1) (kg m² rad s^-1 also acceptable)

Question 48

What is the unit of gravitational potential?

Answer 48

Joules per kilogram (J kg^-1)

Question 49

Shown is the relationship used to find the gravitational potential at a point in space. How could you find the gravitational potential energy of a mass, m, at that point?

Answer 49

Multiply the gravitational potential by the mass: E_p = Vm

Question 50

Derive the expression shown below for escape velocity.

Answer 50

E_k + E_p = 0 ½mv² + (-GMm/r) = 0 ½mv² = GMm/r v² = 2GM/r **v = √2GM/r**

Question 51

What is the unit of Luminosity

Answer 51

Watts (W)

Question 52

What is the unit of apparent brightness

Answer 52

Watts per square metre (Wm^-2)

Question 53

What is meant by the Schwarzschild radius of a black hole?

Answer 53

The Schwarzchild radius of a black hole is a radius of a sphere such that if all the mass of an object were compressed into it, light would not be able to escape from it's surface, thereby forming the black hole.

Question 54

The following experimental setup can be used to determine the moment of inertia of a disc which is allowed to rotate on a frictionless air bearing turntable. Explain the measurments which will need to be taken, and how to use these to determine the moment of inertia of the disc.

Answer 54

Find the **torque** applied to the disc by using the relationship T = Fr. Use the light gate to measure the **angular acceleration** of the disc. Vary the hanging mass to vary the torque, and repeat a few times. Plot T against α, and take the gradient of the straight line which will be equal to the monemt of inertia of the disc.

Question 55

An ice skater is spinning with his arms outward as shown. He then pulls his arms inward. Explain what happens to his angular momentum **and** rotational kinetic energy.

Answer 55

Angular momentum **constant** because angular momentum is always conserved Rotational kinetic energy **increases** because angular velocity increases.

Question 56

What is meant by a conservative field?

Answer 56

The path taken between two points in the field does not affect the work done (energy) used

Question 57

What is meant by the **Schwarzchild radius** of a black hole?

Answer 57

The Schwarzchild radius is the distance from the centre of a black hole to its 'event horizon'.

Question 58

Which two things does the **moment of inertia** of an object depend on?

Answer 58

1. The mass of the object 2. The distribution of mass around a given axis of rotation

Question 59

How would you convert between 1. astronmical units (AU) and metres? 2. light years and metres?

Answer 59

1. 1 AU = 1.5 x 10¹¹ m (given in data sheet) 2. 1 light year = 3 x 10⁸ x (365.25 x 24 x 60 x 60) = **9.46 x 10¹⁵ m** (NOT in data sheet!)