Mechanics Course - 2024-25 Autumn Term

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Questions and Answers

What significant realization was made regarding objects in motion during the 16th century?

Objects naturally come to rest without an external force. (correct)

Planets do not follow the same laws of motion as tennis balls.

The laws of motion are the same for all objects, regardless of size.

Friction plays a crucial role in the motion of celestial bodies.

Which mathematical language was developed to describe change in mechanics?

Calculus (correct)

Geometry

Statistics

Algebra

Which concept is emphasized as central to physics today that originated from mechanics?

Quantum mechanics

Thermodynamics

Conservation laws (correct)

Relativity of time

What limitation of classical mechanics is highlighted in relation to sub-atomic and galactic scales?

Ignoring the roles of quantum effects Signup and view all the answers

How do classical mechanics and quantum mechanics differ at typical room temperatures?

Heavier atoms are almost perfectly described by Newton's laws. Signup and view all the answers

What is a common misconception students might have regarding mechanics?

All motion requires an external force. Signup and view all the answers

Which of the following best summarizes the role of mechanics in physics education?

It provides foundational understanding of physical laws. Signup and view all the answers

What concept, derived from mechanics, is crucial for understanding the motion of both tennis balls and planets?

The principles of inertia and motion Signup and view all the answers

What happens to the position x when time t is much less than tau τ?

x increases linearly with time. Signup and view all the answers

What is the result for the position change x - x0 when time t is much greater than tau τ?

x - x0 equals v0 multiplied by τ. Signup and view all the answers

How is the velocity represented when considering position as a function in the work-energy theorem?

v = v(x) Signup and view all the answers

In the context of different journeys from xi to xf, what may differ despite specifying details of the journey?

The values of force and velocity. Signup and view all the answers

What approximation is made about e^(-t/τ) when t is much greater than τ?

e^(-t/τ) approximates to 0. Signup and view all the answers

When evaluating forces in terms of position in the work-energy theorem, what is the general equation form?

F = F(x) Signup and view all the answers

Which component does not affect the calculation of force and velocity as functions of position?

The final time tf. Signup and view all the answers

What mathematical concept is used to relate changes in velocity and position in the work-energy theorem?

Chain rule. Signup and view all the answers

What is the equation of motion for the internal coordinate related to two masses on a spring?

$, \mu , d¨ = -s(d)$ Signup and view all the answers

How does the concept of reduced mass change when one mass is significantly larger than the other?

The factor of two in the acceleration dissipates. Signup and view all the answers

What happens to the acceleration of the spring length when two equal masses are attached?

It is twice as large as expected. Signup and view all the answers

What is the relevant mass for determining the internal motion of two interacting bodies?

$\mu = \frac{m_1 m_2}{m_1 + m_2}$ Signup and view all the answers

Which of the following statements about collisions is correct?

Collision interactions are brief. Signup and view all the answers

What can be concluded about the potential function U(x) near its minimum at x = x0?

It appears quadratic when sufficiently zoomed in. Signup and view all the answers

Which term in the Taylor series expansion of U(x) is zero at the minimum?

U'(x0) Signup and view all the answers

In what scenario can higher order terms in the Taylor series expansion be ignored?

When x − x0 is very small. Signup and view all the answers

What is the spring constant s defined as in the context of the potential well?

U''(x0) Signup and view all the answers

What does the force F(x) equate to when derived from the potential function near its minimum?

−s(x − x0) Signup and view all the answers

What characteristic do functions need to have to become approximately quadratic near their minimum?

Analytic at x0. Signup and view all the answers

What type of motion can almost all small oscillations be classified as?

Simple harmonic motion. Signup and view all the answers

What happens to the cubic and higher order terms when x is very close to x0?

They can be ignored. Signup and view all the answers

What represents the position of the centre of mass in relation to two masses m1 and m2?

$R = \frac{m_1 \vec{r_1} + m_2 \vec{r_2}}{M}$ Signup and view all the answers

According to the definitions given, what happens if m1 is greater than m2?

The centre of mass is closer to r1. Signup and view all the answers

How does the centre of mass of a two-body system behave under the influence of external forces?

It free falls under the external gravitational force. Signup and view all the answers

What is the equation for total external force acting on the centre of mass?

$F_{ext} = \sum_{i=1}^N \vec{F_i}$ Signup and view all the answers

In which scenario does the centre of mass lie exactly between r1 and r2?

When m1 equals m2. Signup and view all the answers

What factor influences the position of the centre of mass between r1 and r2?

The ratio of the masses m1 and m2. Signup and view all the answers

Which equation correctly represents the motion of the centre of mass under gravitational force?

$F_{ext} = m_1 \vec{g} + m_2 \vec{g}$ Signup and view all the answers

What is the result of subtracting m2$ _1$ from the numerator when evaluating the centre of mass?

It simplifies the calculation of the centre of mass. Signup and view all the answers

What happens to a particle when its energy E is less than the potential U1?

The particle accelerates to the right and decelerates to the left. Signup and view all the answers

In the scenario where U1 < E < U2, what is a key behavior of the particle within the potential well?

The particle oscillates back and forth within the well. Signup and view all the answers

When energy E exceeds U2, what arises regarding the motion of the particle?

The particle has enough energy to overcome the potential barrier. Signup and view all the answers

How can the behavior of a particle in potential U(x) be analogously understood using a ball rolling on a hill?

The gravitational potential energy is equal to U(x). Signup and view all the answers

In the case of small oscillations about equilibrium, what behavior is observed if the energy E is reduced towards U(x0)?

The potential function approaches a quadratic shape. Signup and view all the answers

What is true about the kinetic energy K when E equals U(x)?

K is zero. Signup and view all the answers

What does it mean for a particle to 'bounce off' a potential barrier?

The particle reverses direction after losing all kinetic energy. Signup and view all the answers

When considering the gravitational potential energy analogy, what does h(x) equal for the ball on the hill?

U(x) divided by mg. Signup and view all the answers

Study Notes