Forces (HSC SSCE Physics): Revision Notes

Investigating the Motion of Objects on Inclined Planes

Introduction

Understanding how objects behave on sloped surfaces is fundamental to mechanics. When we analyze inclined planes, we're examining a common real-world scenario where gravity, friction, and support forces interact in interesting ways. This topic builds on our knowledge of forces and Newton's laws, applying them to situations where the surface isn't horizontal.

Forces acting on objects on inclined planes

When an object rests on or moves along an inclined surface, three primary forces act upon it:

• Gravitational force ($F_{\text{Earth on object}}$): This force always acts vertically downward toward the center of Earth, with magnitude $F_{\text{Earth on object}} = mg$, where $m$ is the object's mass and $g = 9.8 \, \text{N kg}^{-1}$ is the acceleration due to gravity.
• Normal force ($F_{\text{surface on object, normal}}$): This force acts perpendicular to the surface of the slope, pushing the object away from the surface. The normal force prevents the object from sinking into the inclined plane.
• Friction force ($F_{\text{surface on object, friction}}$): This force acts parallel to the slope's surface. When an object is stationary or moving up the slope, friction acts down the slope. When an object slides down, kinetic friction opposes the motion by acting up the slope.

These three forces are illustrated in the diagram below, which shows an object on an inclined plane at angle $\theta$ to the horizontal:

Choosing an appropriate coordinate system

In most physics problems, we use horizontal and vertical coordinate axes. However, when dealing with inclined planes, selecting axes aligned with the slope makes our calculations much simpler. Here's why:

We align the x-axis parallel to the slope and the y-axis perpendicular to the slope. This choice is strategic because:

• The object can only accelerate along the slope (parallel direction), not into or away from it
• If the object cannot move perpendicular to the slope, the net force in that direction must be zero
• Two of our three forces (normal force and friction) already act along these axes
• Only the gravitational force needs to be decomposed into components

Decomposing the gravitational force

While the normal force acts along the y-axis and friction acts along the x-axis, the gravitational force acts vertically downward. We must decompose it into components:

• Parallel component (along x-axis): $F_{\text{Earth on object, x}} = F_{\text{Earth on object}} \sin\theta = mg\sin\theta$This component acts down the slope, trying to make the object slide downward.
• Perpendicular component (along y-axis): $F_{\text{Earth on object, y}} = F_{\text{Earth on object}} \cos\theta = mg\cos\theta$This component acts into the slope, compressing the object against the surface.

Newton's second law in component form

We can now apply Newton's second law separately in each direction.

Perpendicular direction (y-axis)

Since the object cannot accelerate perpendicular to the slope, the net force in this direction equals zero:

$F_{\text{net}, y} = \Sigma F_y = F_{\text{surface on object, normal}} - F_{\text{Earth on object, y}} = 0$

Expanding this:

$F_{\text{surface on object, normal}} - F_{\text{Earth on object}} \cos\theta = 0$

Therefore:

$F_{\text{surface on object, normal}} = F_{\text{Earth on object}} \cos\theta = mg\cos\theta$

This tells us the normal force equals the perpendicular component of the gravitational force.

Parallel direction (x-axis)

In the parallel direction, the object may or may not accelerate, depending on whether friction can balance the gravitational component:

$F_{\text{net}, x} = \Sigma F_x = F_{\text{surface on object, friction}} - F_{\text{Earth on object, x}} = m_{\text{object}} a$

Expanding this:

$F_{\text{surface on object, friction}} - F_{\text{Earth on object}} \sin\theta = m_{\text{object}} a$

If $a = 0$, the object is in equilibrium. If $a \neq 0$, the object accelerates down the slope.

Static equilibrium on inclined planes

When an object remains stationary on a slope, it is in static equilibrium. This means the net force in both directions equals zero. We've already seen that the net force perpendicular to the slope is zero. For the parallel direction:

$F_{\text{net}, x} = 0$

Therefore:

$F_{\text{surface on object, friction}} = F_{\text{Earth on object}} \sin\theta = mg\sin\theta$

This equation tells us that the static friction force must exactly balance the gravitational component pulling the object down the slope. This equilibrium can only exist if the maximum possible static friction force is greater than or equal to $mg\sin\theta$.

Sliding motion on inclined planes

When an object slides down an inclined plane, the kinetic friction force is less than the gravitational component down the slope. In this case, there is a net force parallel to the slope:

$F_{\text{net}, x} = F_{\text{surface on object, friction}} - F_{\text{Earth on object}} \sin\theta = m_{\text{object}} a$

Rearranging to find acceleration:

$a = \frac{F_{\text{surface on object, friction}} - F_{\text{Earth on object}} \sin\theta}{m_{\text{object}}}$

Or equivalently:

$a = \frac{F_{\text{surface on object, friction}} - mg\sin\theta}{m}$

This acceleration is directed down the slope. By measuring this acceleration experimentally, we can calculate the kinetic friction force acting between the object and the surface.

Critical angle

The critical angle ($\theta_c$) is the angle at which an object just begins to slide down the inclined plane. At this angle, the static friction force reaches its maximum possible value:

$F_{\text{static friction max}} = m_{\text{object}} g \sin\theta_c$

By adjusting the angle of an inclined plane until an object just starts to slide, we can measure the critical angle and calculate the maximum static friction force between the surfaces. This is a useful experimental technique for studying friction between different materials.

Worked example: Harriet on the slide

Investigation: Motion of objects on inclined planes

Measuring the acceleration of objects on inclined planes allows us to determine the frictional forces acting between different surfaces. This investigation can also help us find the maximum static friction force by identifying the critical angle at which objects begin to slide.

Aim

Design your own investigation using objects sliding on inclined planes. For example, you might investigate:

• How does the angle of the incline affect the acceleration of a sliding object?
• What is the relationship between surface materials and friction on an incline?
• How does mass affect the motion of objects on an inclined plane?

Write a clear aim statement and formulate an inquiry question or hypothesis to guide your investigation.

Materials

Essential equipment:

• An inclined plane (adjustable to different angles is ideal)
• One or more objects to slide down the inclined plane
• A protractor or angle-measuring device

Additional equipment (depending on your investigation):

• Timing equipment (stopwatch, light gates, or motion sensors)
• Different surface materials to test various friction coefficients
• Various masses or weights
• Measuring tape or ruler for distance measurements
• Video recording equipment for motion analysis

Risk assessment

Method

Develop a detailed, step-by-step method for your investigation. Your method should:

• Clearly state what you will measure and how
• Identify your independent, dependent, and controlled variables
• Explain how you will minimise uncertainties in measurements
• Include sufficient detail that another student could replicate your experiment
• Specify the number of trials or measurements you will take

Have your teacher check and approve your method before you begin collecting data.

Results

Record all measurements as you make them, including:

• Appropriate units for all quantities
• Estimated uncertainties in your measurements
• Multiple trials to assess repeatability

Organise your results in a clear table format. For example:

Angle ($\theta$)	Trial 1 time (s)	Trial 2 time (s)	Trial 3 time (s)	Average time (s)	Acceleration (m $\cdot$ s$^{-2}$)
...	...	...	...	...	...

Analysis of results

Analyze your data to answer your inquiry question or test your hypothesis. This might involve:

• Calculating accelerations from distance and time measurements
• Determining friction forces using $F_{\text{friction}} = mg\sin\theta - ma$
• Plotting graphs to identify relationships between variables
• Calculating percentage differences or uncertainties
• Comparing calculated values with theoretical predictions

Show all calculations clearly and explain your reasoning.

Discussion

Reflect on your investigation:

• Did you answer your inquiry question? Was your hypothesis supported by the data?
• If other groups conducted similar investigations, do their results agree with yours? If not, why might they differ?
• What were the main sources of uncertainty in your measurements?
• How could you improve the investigation if you were to repeat it?
• What assumptions did you make in your analysis? Are they valid?

Conclusion

Write a concise conclusion that:

• Summarizes the key findings of your investigation
• Directly addresses your aim and inquiry question
• States any quantitative results with appropriate units and uncertainties
• Identifies the most significant limitation of your investigation

Remember!