Applications: Objects in Motion — JEE Advanced Physics

Key Concepts

Key concepts will be added here.

Examples

Examples will be added here.

Introduction to Applications: Objects in Motion

Welcome to Applications: Objects in Motion. Content to be added.

Motion on Inclined Planes

Why Tilted Coordinates?

For objects moving along an inclined plane, it's almost always more convenient to choose a tilted coordinate system:

x-axis: Parallel to the incline
y-axis: Perpendicular to the incline

This choice is strategic because acceleration is usually along the incline (x-axis), simplifying calculations.

Block on inclined plane

Coordinate system for incline

Resolving Weight on an Incline

Weight $\vec{W} = m\vec{g}$ always acts vertically downward. On an incline at angle $\theta$ , we resolve it:

Weight resolution on incline

Parallel to incline (down the slope):

$W_{\parallel} = mg\sin\theta$

Perpendicular to incline (into the surface):

$W_{\perp} = mg\cos\theta$

Remember: $mg\sin\theta$ pulls down the slope, $mg\cos\theta$ presses into the surface

Normal Force on Incline

The normal force $N$ acts perpendicular to the incline surface. For a block on an incline with no perpendicular acceleration:

$N = mg\cos\theta$

Note: $N < mg$ (the full weight) because only the perpendicular component of weight is balanced by the normal force.

Example 1: Sliding Down Smooth Incline

A block starts from rest on a smooth (frictionless) incline at $\theta = 45°$ . What is its speed after 2 seconds?

Block released on incline

FBD of block on incline

Solution: Only force along incline is $mg\sin\theta$

$\sum F_x = ma \implies mg\sin\theta = ma$

$a = g\sin45° = 10 \times \dfrac{1}{\sqrt{2}} = 5\sqrt{2} \, \text{m/s}^2$

Using kinematics: $u = 0$ , $a = 5\sqrt{2}$ , $t = 2$ s

$v = u + at = 0 + (5\sqrt{2})(2) = 10\sqrt{2} \approx 14.14 \, \text{m/s}$

If the incline angle were increased to 60°, would the acceleration be greater or smaller than at 45°?

Example 2: Block Pulled Up Incline

A block on a smooth incline ( $\theta = 30°$ ) is pulled up by a string parallel to the incline with tension $T = 60$ N. Mass = 5 kg. Find acceleration.

Block pulled up incline

FBD of block

Solution: Along incline (up positive):

$\sum F_x = T - mg\sin\theta = ma$

$60 - (5)(10)(0.5) = 5a$

$60 - 25 = 5a \implies a = 7 \, \text{m/s}^2 \text{ (up incline)}$

Example 3: Incline with Friction

A 4 kg block on a rough incline ( $\theta = 37°$ , $\mu_k = 0.25$ ) slides down. Find acceleration.

Solution:

Normal: $N = mg\cos37° = 40 \times 0.8 = 32$ N

Friction (opposes motion, up incline): $f_k = \mu_k N = 0.25 \times 32 = 8$ N

Along incline (down positive):

$mg\sin37° - f_k = ma$

$(40)(0.6) - 8 = 4a$

$24 - 8 = 4a \implies a = 4 \, \text{m/s}^2$

On an inclined plane, does friction reduce the acceleration compared to a smooth (frictionless) incline?

Problem-Solving Strategies and Practice

Systematic Problem-Solving Approach

Follow these steps for dynamics problems:

Identify the system: What object(s) are you analyzing?
Draw FBDs: For each object, show all external forces
Choose coordinates: Align axes with acceleration when possible
Apply Newton's Second Law: $\sum \vec{F} = m\vec{a}$ in component form
Solve equations: Use algebra or simultaneous equations
Check: Do signs make sense? Are units correct?

Practice 1: Hanging Lamp

A 2 kg lamp hangs motionless from a wire. Find tension.

Lamp hanging

FBD of lamp

Solution: Equilibrium ( $a_y = 0$ )

Forces: Weight $W = mg = 20$ N (down), Tension $T$ (up)

$\sum F_y = 0 \implies T - W = 0 \implies T = 20 \, \text{N}$

Practice 2: Block Pushed on Surface

A 10 kg block on smooth horizontal floor pushed with 20 N horizontally. Find normal force and acceleration.

Block pushed on surface

FBD of pushed block

Solution:

Vertical: Equilibrium → $N = W = 100$ N

Horizontal: $F = ma \implies 20 = 10a \implies a = 2 \, \text{m/s}^2$

Normal force = 100 N, acceleration = 2 m/s² (horizontal push doesn't affect vertical forces!).

In the pushed block example, does the 20 N horizontal push affect the normal force?

Practice 3: Block Against Wall

A 2 kg block held against smooth vertical wall by 30 N horizontal force and vertical string. Find normal force and tension for equilibrium.

Block against wall

FBD of block

Solution: Equilibrium ( $a_x = 0$ , $a_y = 0$ )

Horizontal: $F - N = 0 \implies N = 30$ N

Vertical: $T - W = 0 \implies T = mg = 20$ N

Practice 4: Weighing Machine in Elevator

An 80 kg person on a scale in an elevator. Find scale reading when elevator:

Person in elevator

FBD of person

(a) Moving at constant 2 m/s upward:

$a = 0$ → $N = W = 800$ N → Reading = 80 kg

(b) Accelerating upward at 2 m/s²:

$N - W = ma$ → $N = 800 + 160 = 960$ N → Reading = 96 kg

(c) Accelerating downward at 2 m/s²:

$N - W = m(-2)$ → $N = 800 - 160 = 640$ N → Reading = 64 kg

Does a person feel heavier (higher scale reading) when an elevator accelerates upward?

Practice 5: Force on Pulley Support

An Atwood machine with $m_1 = 4$ kg, $m_2 = 6$ kg has tension $T = 48$ N. What force does the clamp exert on the pulley axle?

Solution: Pulley is massless and stationary (equilibrium)

Forces on pulley:

Tension $T = 48$ N downward (from $m_1$ side)
Tension $T = 48$ N downward (from $m_2$ side)
Support force $F_{clamp}$ upward

$F_{clamp} - T - T = 0 \implies F_{clamp} = 2T = 96 \, \text{N}$

The clamp exerts 96 N upward on the pulley.

Key Problem-Solving Tips

Always start with a clear FBD
Be consistent with sign conventions
For connected systems, identify constraints (same acceleration, etc.)
Check if object is in equilibrium or accelerating
Remember: $N \neq mg$ in general!
Friction is self-adjusting (static) or constant (kinetic)
Pseudo forces appear only in non-inertial frames

Key Concepts

Key concepts will be added here.

Examples

Examples will be added here.

Introduction to Applications: Objects in Motion

Welcome to Applications: Objects in Motion. Content to be added.

Motion on Inclined Planes

Why Tilted Coordinates?

For objects moving along an inclined plane, it's almost always more convenient to choose a tilted coordinate system:

x-axis: Parallel to the incline
y-axis: Perpendicular to the incline

This choice is strategic because acceleration is usually along the incline (x-axis), simplifying calculations.

Block on inclined plane

Coordinate system for incline

Resolving Weight on an Incline

Weight $\vec{W} = m\vec{g}$ always acts vertically downward. On an incline at angle $\theta$ , we resolve it:

Weight resolution on incline

Parallel to incline (down the slope):

$W_{\parallel} = mg\sin\theta$

Perpendicular to incline (into the surface):

$W_{\perp} = mg\cos\theta$

Remember: $mg\sin\theta$ pulls down the slope, $mg\cos\theta$ presses into the surface

Normal Force on Incline

The normal force $N$ acts perpendicular to the incline surface. For a block on an incline with no perpendicular acceleration:

$N = mg\cos\theta$

Note: $N < mg$ (the full weight) because only the perpendicular component of weight is balanced by the normal force.

Example 1: Sliding Down Smooth Incline

A block starts from rest on a smooth (frictionless) incline at $\theta = 45°$ . What is its speed after 2 seconds?

Block released on incline

FBD of block on incline

Solution: Only force along incline is $mg\sin\theta$

$\sum F_x = ma \implies mg\sin\theta = ma$

$a = g\sin45° = 10 \times \dfrac{1}{\sqrt{2}} = 5\sqrt{2} \, \text{m/s}^2$

Using kinematics: $u = 0$ , $a = 5\sqrt{2}$ , $t = 2$ s

$v = u + at = 0 + (5\sqrt{2})(2) = 10\sqrt{2} \approx 14.14 \, \text{m/s}$

If the incline angle were increased to 60°, would the acceleration be greater or smaller than at 45°?

Example 2: Block Pulled Up Incline

A block on a smooth incline ( $\theta = 30°$ ) is pulled up by a string parallel to the incline with tension $T = 60$ N. Mass = 5 kg. Find acceleration.

Block pulled up incline

FBD of block

Solution: Along incline (up positive):

$\sum F_x = T - mg\sin\theta = ma$

$60 - (5)(10)(0.5) = 5a$

$60 - 25 = 5a \implies a = 7 \, \text{m/s}^2 \text{ (up incline)}$

Example 3: Incline with Friction

A 4 kg block on a rough incline ( $\theta = 37°$ , $\mu_k = 0.25$ ) slides down. Find acceleration.

Solution:

Normal: $N = mg\cos37° = 40 \times 0.8 = 32$ N

Friction (opposes motion, up incline): $f_k = \mu_k N = 0.25 \times 32 = 8$ N

Along incline (down positive):

$mg\sin37° - f_k = ma$

$(40)(0.6) - 8 = 4a$

$24 - 8 = 4a \implies a = 4 \, \text{m/s}^2$

On an inclined plane, does friction reduce the acceleration compared to a smooth (frictionless) incline?

Problem-Solving Strategies and Practice

Systematic Problem-Solving Approach

Follow these steps for dynamics problems:

Identify the system: What object(s) are you analyzing?
Draw FBDs: For each object, show all external forces
Choose coordinates: Align axes with acceleration when possible
Apply Newton's Second Law: $\sum \vec{F} = m\vec{a}$ in component form
Solve equations: Use algebra or simultaneous equations
Check: Do signs make sense? Are units correct?

Practice 1: Hanging Lamp

A 2 kg lamp hangs motionless from a wire. Find tension.

Lamp hanging

FBD of lamp

Solution: Equilibrium ( $a_y = 0$ )

Forces: Weight $W = mg = 20$ N (down), Tension $T$ (up)

$\sum F_y = 0 \implies T - W = 0 \implies T = 20 \, \text{N}$

Practice 2: Block Pushed on Surface

A 10 kg block on smooth horizontal floor pushed with 20 N horizontally. Find normal force and acceleration.

Block pushed on surface

FBD of pushed block

Solution:

Vertical: Equilibrium → $N = W = 100$ N

Horizontal: $F = ma \implies 20 = 10a \implies a = 2 \, \text{m/s}^2$

Normal force = 100 N, acceleration = 2 m/s² (horizontal push doesn't affect vertical forces!).

In the pushed block example, does the 20 N horizontal push affect the normal force?

Practice 3: Block Against Wall

A 2 kg block held against smooth vertical wall by 30 N horizontal force and vertical string. Find normal force and tension for equilibrium.

Block against wall

FBD of block

Solution: Equilibrium ( $a_x = 0$ , $a_y = 0$ )

Horizontal: $F - N = 0 \implies N = 30$ N

Vertical: $T - W = 0 \implies T = mg = 20$ N

Practice 4: Weighing Machine in Elevator

An 80 kg person on a scale in an elevator. Find scale reading when elevator:

Person in elevator

FBD of person

(a) Moving at constant 2 m/s upward:

$a = 0$ → $N = W = 800$ N → Reading = 80 kg

(b) Accelerating upward at 2 m/s²:

$N - W = ma$ → $N = 800 + 160 = 960$ N → Reading = 96 kg

(c) Accelerating downward at 2 m/s²:

$N - W = m(-2)$ → $N = 800 - 160 = 640$ N → Reading = 64 kg

Does a person feel heavier (higher scale reading) when an elevator accelerates upward?

Practice 5: Force on Pulley Support

An Atwood machine with $m_1 = 4$ kg, $m_2 = 6$ kg has tension $T = 48$ N. What force does the clamp exert on the pulley axle?

Solution: Pulley is massless and stationary (equilibrium)

Forces on pulley:

Tension $T = 48$ N downward (from $m_1$ side)
Tension $T = 48$ N downward (from $m_2$ side)
Support force $F_{clamp}$ upward

$F_{clamp} - T - T = 0 \implies F_{clamp} = 2T = 96 \, \text{N}$

The clamp exerts 96 N upward on the pulley.

Key Problem-Solving Tips

Always start with a clear FBD
Be consistent with sign conventions
For connected systems, identify constraints (same acceleration, etc.)
Check if object is in equilibrium or accelerating
Remember: $N \neq mg$ in general!
Friction is self-adjusting (static) or constant (kinetic)
Pseudo forces appear only in non-inertial frames