Midterm practice

Forces and Newton's Laws of Motion

Net Force on an Object

Understanding the net force acting on an object is fundamental to Newton's laws. The net force is the vector sum of all forces acting on the object.

• Definition: The net force ($\vec{F}_{\text{net}}$) is the sum of all individual forces acting on an object.

• Example: For three stacked books with weights 3 N, 5 N, and 10 N, the net force on the middle book (Y) is the difference between the upward normal force and the downward forces (its own weight and the weight of the book above).

• Equation: $\vec{F}_{\text{net}} = \sum \vec{F}_i$

Work Done by Forces

Work is the energy transferred to or from an object via the application of force along a displacement.

• Work by a Person: Positive if the force and displacement are in the same direction.

• Work by Gravity: Positive if the object moves in the direction of gravity, negative if against.

• Equation: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

• Example: Pushing a box down a ramp: work by the girl is positive, work by gravity is positive.

Motion in a Plane and Circular Motion

Acceleration in Circular Motion

When an object moves in a circle at constant speed, its acceleration is always directed toward the center of the circle (centripetal acceleration).

• Direction: Always perpendicular to the velocity.

• Equation: $a_c = \frac{v^2}{r}$

• Key Point: The magnitude of acceleration is constant, but its direction changes continuously.

Conservation of Energy

Work and Energy on an Inclined Plane

When a block slides down an inclined plane, energy conservation and friction must be considered.

• Potential Energy: $U = mgh$

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

• Work by Friction: $W_{\text{friction}} = -\mu_k mg\cos\theta \cdot d$

• Equation for Final Speed: $v = \sqrt{2g d (\sin\theta - \mu_k \cos\theta)}$

• Example: Block sliding down a ramp with friction.

Rotational Dynamics and Torque

Torque and Rotational Motion

Torque is the rotational equivalent of force and causes angular acceleration.

• Definition: Torque ($\tau$) is given by $\tau = rF\sin\theta$

• Example: Calculating the torque due to gravity for a mass attached to a rod at an angle.

• Application: Used to analyze systems like swings, rods, and human limbs.

Equilibrium and Tension

Objects in equilibrium have zero net force and zero net torque.

• Key Point: The tension in a string or cable depends on the angle and the forces acting on the object.

• Example: A stick attached to a wall and held by a string; the smallest tension occurs when the stick is vertical.

Oscillations and Springs

Hooke's Law and Spring Compression

Springs obey Hooke's Law, which relates the force exerted by a spring to its displacement from equilibrium.

• Equation: $F_{\text{spring}} = -k \Delta x$

• Application: Used to analyze systems like shower curtain rods and masses on springs.

• Energy Stored: $U = \frac{1}{2}k(\Delta x)^2$

Friction and Energy Dissipation

When a mass slides on a surface with friction, energy is dissipated as heat.

• Coefficient of Kinetic Friction: $\mu_k$

• Equation: $f_k = \mu_k N$

• Example: Determining $\mu_k$ from energy considerations as a mass moves on a spring and comes to rest.

Applications: Human Biomechanics and Therapy

Torque in Human Limbs

Biomechanical analysis often involves calculating the torque produced by muscles and external forces.

• Example: Calculating the maximum force the biceps can withstand when holding a ball at a certain angle.

• Equation: $\tau = rF\sin\theta$

Summary Table: Work Done by Various Forces

Force	Sign of Work	Explanation
Gravity	Positive/Negative	Depends on direction of displacement relative to gravity
Friction	Negative	Always opposes motion
Applied Force	Positive/Negative	Depends on direction of force and displacement

Additional info:

• Some diagrams and questions refer to real-world applications, such as aquatic therapy and trust falls, which are analyzed using principles of torque, energy, and equilibrium.

• All problems are based on core physics topics: forces, energy, circular motion, rotational dynamics, and oscillations.