University Physics: Motion, Forces, and Energy Study Guide

Basics of Motion

• Speed is defined as the distance travelled per unit time.

• The formula for speed is:     $\text{speed} = \frac{\text{distance travelled}}{\text{time}}$

• The units for speed are $\text{m/s}$.

• Velocity is defined as the speed in a given direction.

• The formula for velocity is:     $\text{velocity} = \frac{\text{displacement}}{\text{time}}$     $v = \frac{s}{t}$

• The units for velocity are $\text{m/s}$.

• If the speed of an object is changing, the object is either accelerating or decelerating.

• Average speed is used when speed is changing throughout a journey:     $\text{average speed} = \frac{\text{total distance travelled}}{\text{total time}}$

• Acceleration is defined as the rate of change of velocity.

• The formula for acceleration is:     $\text{acceleration} = \frac{\text{change in velocity}}{\text{change in time}}$     $a = \frac{\Delta v}{\Delta t}$

• The units for acceleration are $\text{m/s}^2$.

• Deceleration is defined as a negative acceleration.

• Before performing calculations, ensure units are equivalent. Standard conversions involve:

  • Distance/Displacement: $\text{mm}$, $\text{cm}$, $\text{m}$, or $\text{km}$.

  • Time: $\text{ms}$, $\text{s}$, $\text{minutes}$, or $\text{hours}$.

  • Most answers should be standardized to $\text{metres}$ and $\text{seconds}$.

Interpreting Displacement-Time Graphs

• The gradient of a displacement-time graph represents the velocity (on a distance-time graph, it represents speed).

• An object is at rest when the gradient is horizontal (flat).

• An object is moving at a constant speed or velocity when the gradient is straight (linear).

• An object is accelerating when the line is curved and the gradient is increasing.

• An object is decelerating when the line is curved and the gradient is decreasing.

• A negative gradient indicates the object is returning to its starting point.

• To calculate speed or velocity from the graph:     $\text{speed or velocity} = \text{gradient} = \frac{\text{change in } Y}{\text{change in } x}$

• If this is calculated for a curved section, the result is the average speed.

Interpreting Velocity-Time Graphs

• The gradient of a velocity-time graph represents acceleration.

• An object is at rest when the speed or velocity is zero.

• An object is moving at a constant speed when the line is horizontal (gradient is zero).

• An object is accelerating when the line has a positive gradient.

• An object is decelerating when the line has a negative gradient.

• An object is moving with constant acceleration when the line is straight.

• An object is moving with changing acceleration when the line is curved.

• To calculate acceleration from the graph:     $\text{acceleration} = \text{gradient} = \frac{\text{change in } y}{\text{change in } x}$

• The area under the velocity-time graph represents the total distance travelled.

• To calculate the area, the space under the graph can be split into geometric shapes such as rectangles and triangles.

Gravity and Objects in Free Fall

• The acceleration of free fall, denoted as $g$, is approximately $9.8\,m/s^2$.

• In a uniform gravitational field and in the absence of air or liquid resistance, all objects fall with this constant acceleration of $9.8\,m/s^2$.

• In the presence of air or liquid resistance, objects fall with decreasing acceleration:

  • Initially, there is no air resistance, and the only force is weight.

  • As the object accelerates, its speed increases, which in turn increases air resistance.

  • The increasing air resistance reduces the resultant downward force, causing acceleration to decrease.

  • Eventually, weight and air resistance become equal and opposite.

  • At this stage, there is no resultant force and no acceleration.

  • The object reaches a constant speed known as terminal velocity.

Extension and Hooke's Law

• Forces can change the size and shape of an object.

• Elastic solids extend when force is applied and return to their original shape/size when the force is removed.

• The spring constant ($k$) is the force required per unit of extension.

• Hooke's Law formula:     $\text{spring constant} = \frac{\text{force applied}}{\text{extension}}$     $k = \frac{F}{x}$

• The units for the spring constant are $\text{N/m}$.

• Experimental investigation of extension:

  • Measure the initial length of the object with a ruler.

  • Attach masses incrementally to apply force.

  • Measure and record the new length after each mass is added.

  • Calculate extension by subtracting the initial length from the new length ($extension = new\,length - initial\,length$).

  • Repeat the experiment three times to find an average extension for each mass.

  • Calculate force (weight) by multiplying mass by the gravitational field strength ($9.8\,N/kg$).

• Load-extension graphs:

  • Graphs should be linear and pass through the origin for elastic objects.

  • The gradient of the linear section equals the spring constant ($k$).

  • The point where the graph ceases to be linear is the limit of proportionality; beyond this point, the object stretches irreversibly and the equation $k = \frac{F}{x}$ no longer holds true.

Resultant Forces and Newton's Laws

• A resultant force is a single force describing the combined action of all forces on an object.

• Finding resultant forces along the same straight line:     $\text{Resultant force} = (\text{sum of all forces in one direction}) - (\text{sum of forces in the opposite direction})$

• If forces balance out, the resultant force is zero.

• Newton's First Law: Without a resultant force, an object remains at rest or continues in a straight line at a constant velocity.

• With a resultant force, an object's velocity changes (acceleration) through a change in speed or direction.

• Newton's Second Law: Acceleration is proportional to the resultant force and inversely proportional to mass.     $\text{force} = \text{mass} \times \text{acceleration}$     $F = ma$

Circular Motion and Friction

• A resultant force is required for circular motion because the object is always changing direction (and thus velocity).

• This force must act perpendicular to the direction of motion (e.g., gravity acting on an orbiting body).

• Relations in circular motion:

  • If mass and radius are constant: Increasing force increases speed.

  • If mass and speed are constant: Increasing force decreases radius.

  • If mass increases: A higher force is required to maintain constant speed and radius.

• Friction (drag) is a force between surfaces that impedes motion and causes heating.

• Friction can occur in liquids or gases (e.g., air resistance).

The Turning Effect of Forces (Moments)

• The pivot point is the point about which an object rotates.

• Rotation occurs if the force applied is in a different line to the pivot.

• If the force is not perpendicular to the object, trigonometry is used to find the perpendicular distance to the pivot.

• A moment is the measure of the turning effect of a force.

• Moment formula:     $\text{moment of a force} = \text{force} \times \text{perpendicular distance}$     $\text{moment} = Fd$

• Example: A bike pedal arm turns about a pivot when the foot applies force.

• Equilibrium occurs when the clockwise moment equals the anticlockwise moment, resulting in no overall rotation.

• Example: A see-saw is balanced only if the moments created by each person's weight are equal.

• Experiment for Equilibrium:

  • Pivot a uniform ruler at its center.

  • Place different masses on either side until balanced.

  • Calculate forces ($weight = mass \times 9.18\,N/kg$).

  • Verify that $\text{anticlockwise moments} = \text{clockwise moments}$.

Centre of Gravity and Stability

• The centre of gravity is the point at which all of an object's weight is considered to act.

• Finding the centre of gravity of an irregularly shaped plane lamina:

  • Hang the lamina and a plumb line (thread) from the same point.

  • Mark the path of the plumb line.

  • Repeat the process from different suspension points.

  • The intersection point of the marked lines is the centre of gravity.

• Stability factors:

  • Stable Equilibrium: The centre of mass is below the suspension point (e.g., a hanging plant pot).

  • Unstable Equilibrium: The centre of mass is above the suspension point (e.g., a pencil balanced on its point).

  • Toppling: Occurs if the line of action of the weight moves outside the object's base, creating a resultant moment.

  • Objects with a lower centre of gravity and a wider base are more stable.

Momentum and Impulse

• Momentum is the product of mass and velocity:     $p = mv$

• The units for momentum are $\text{kg m/s}$.

• Impulse is the product of force and the time during which it acts, which equals the change in momentum:     $\text{impulse} = \text{force} \times \text{time} = \Delta(mv)$

• The units for impulse are $\text{Ns}$.

• Principle of the Conservation of Momentum: In a collision, total momentum before equals total momentum afterwards.

• Example Recoil Calculation:

  • Scenario: A stationary $10\,kg$ gun fires a $10\,g$ ($0.01\,kg$) bullet at $100\,m/s$.

  • $\text{Total momentum before} = 0$

  • $\Delta(mv) = 0$

  • $0\,kg\,m/s = (0.01\,kg \times 100\,m/s) + (10\,kg \times v)$

  • $1 + 10v = 0$

  • $v = -0.1\,m/s$

  • Recoil speed is the magnitude: $0.1\,m/s$.

Energy Stores and Transfers

• Energy is always conserved: total energy before equals total energy after.

• Common energy stores include kinetic, gravitational potential, chemical, elastic (strain), nuclear, electrostatic, and internal (thermal).

• Kinetic Energy calculation:     $E_k = \frac{1}{2}mv^2$

• Gravitational Potential Energy calculation:     $E_p = mgh$

• The unit for energy is the Joule ($J$).

• Energy Transfer Mechanisms:

  • Forces: e.g., gravity accelerating an object downwards to gain kinetic energy.

  • Electrical currents: e.g., current powering a lamp to emit light and heat.

  • Heating: e.g., a fire heating an object.

  • Waves: e.g., sound waves travelling through air via vibrations.