Systems of Objects with Friction: Videos & Practice Problems

In systems with connected objects experiencing friction, each object's friction must be considered. When analyzing such systems, remember that connected objects share the same velocity and acceleration. Begin by drawing free body diagrams for each object, identifying forces like tension, normal force, and kinetic friction. Use Newton's second law, F = m × a, to set up equations for each object, allowing you to solve for unknowns like acceleration and tension through substitution or addition of equations.

Connected Objects With Friction

Understanding friction in systems with multiple connected objects is crucial for solving physics problems effectively. When dealing with such systems, it is essential to recognize that all connected objects share the same velocity and acceleration, meaning they move together as a single unit. This principle simplifies the analysis of forces acting on each object, especially when friction is involved.

To illustrate this, consider a scenario with two blocks, labeled A and B, where block A has a mass of 5 kg and block B has a mass of 10 kg. The coefficients of kinetic friction are provided, and the goal is to determine the acceleration of the system. The first step involves drawing free body diagrams for both blocks, which helps visualize the forces acting on each. For block A, the forces include the gravitational force (weight), the normal force, the tension force acting in the direction of motion, and the frictional force opposing the motion. Since the blocks are in motion, we specifically deal with kinetic friction.

For block B, the forces include the applied force (90 N), the tension force acting in the opposite direction, the normal force, and the kinetic friction force. The frictional force can be expressed as \( f_k = \mu_k \cdot N \), where \( \mu_k \) is the coefficient of kinetic friction and \( N \) is the normal force. In horizontal motion, the normal force equals the weight of the block, which is calculated as \( N = m \cdot g \), where \( g \) is the acceleration due to gravity (approximately 9.8 m/s²).

To find the acceleration, we apply Newton's second law, \( F = m \cdot a \), for each block. For block A, the equation can be set up as:

\( T - f_k = m_A \cdot a \)

Substituting the expression for kinetic friction gives:

\( T - \mu_k \cdot m_A \cdot g = m_A \cdot a \)

For block B, the equation becomes:

\( F - T - f_k = m_B \cdot a \)

Again substituting for kinetic friction yields:

\( 90 - T - \mu_k \cdot m_B \cdot g = m_B \cdot a \)

By substituting the known values into these equations, we can express the forces in terms of the known masses and the coefficient of friction. After simplifying, we can set up a system of equations to solve for the unknowns, which are the tension and the acceleration. By adding or substituting these equations, we can eliminate the tension variable and solve for the acceleration of the system.

In this example, after performing the calculations, the acceleration of the system is found to be approximately 3.06 m/s². This systematic approach of analyzing forces, drawing free body diagrams, and applying Newton's laws is essential for solving problems involving friction and connected objects in physics.

Two blocks are connected by a cord over a pulley. Block A rests on a rough tabletop. Block B has mass mB=2kg and hangs over the edge of the table. The coefficients of friction between Block A and the tabletop are μs=0.6 and μk=0.4. What is the minimum mass Block A can have to keep the system from starting to move?

• A. 3.33 kg
• B. 5 kg
• C. 32.7 kg

Answer: (A)

Here’s what students ask on this topic:

How do you draw free body diagrams for systems of objects with friction?

To draw free body diagrams for systems of objects with friction, follow these steps:

1. Identify each object in the system and draw them separately.

2. For each object, draw vectors representing all forces acting on it. These include:

• Gravitational force (${\mathrm{mg}}_i$), where $i$ is the object index.
• Normal force ($N_i$).
• Tension force ($T$), if objects are connected by a string or cable.
• Frictional force ($f_k$), which opposes the direction of motion.

3. Label each force clearly and indicate the direction of motion or applied forces.

4. Ensure that the forces are balanced according to Newton's second law, $F=ma$.

What is the role of kinetic friction in systems of connected objects?

Kinetic friction plays a crucial role in systems of connected objects by opposing the motion of each object. It is calculated using the formula:

$f_k={\mu }_k{N}_i$

where ${\mu }_k$ is the coefficient of kinetic friction and $N_i$ is the normal force on the object. Kinetic friction affects the net force acting on each object, influencing the system's overall acceleration. When solving problems, you must account for kinetic friction for each object to accurately determine the system's behavior.

How do you solve for acceleration in systems of objects with friction?

To solve for acceleration in systems of objects with friction, follow these steps:

1. Draw free body diagrams for each object, identifying all forces.

2. Write Newton's second law for each object: $F=ma$.

3. Include frictional forces using $f_k={\mu }_k{N}_i$.

4. Set up equations for each object, considering tension and other forces.

5. Solve the system of equations using substitution or addition to find the unknowns, such as acceleration and tension.

6. Ensure the acceleration is consistent across all objects, as they move together.

What are the common mistakes to avoid when analyzing systems of objects with friction?

Common mistakes to avoid include:

• Forgetting to include all forces in the free body diagrams, such as tension or friction.
• Incorrectly calculating the normal force, which affects the frictional force.
• Assuming static friction when the problem specifies kinetic friction.
• Not considering that connected objects share the same acceleration.
• Failing to correctly apply Newton's second law to each object.
• Ignoring the direction of forces, leading to sign errors in equations.

Carefully follow the steps and double-check your work to avoid these errors.

How do you determine the type of friction (static or kinetic) in a problem involving systems of objects?

To determine the type of friction in a problem involving systems of objects, consider the following:

1. **Problem Statement**: Check if the problem explicitly states whether the objects are moving or stationary.

2. **Motion Status**: If the objects are moving, kinetic friction is involved. If they are stationary but on the verge of moving, static friction applies.

3. **Force Analysis**: Compare the applied force to the maximum static friction force ($f_s={\mu }_s{N}_i$). If the applied force exceeds this value, the objects will move, and kinetic friction will take over.

4. **Given Coefficients**: Use the given coefficients of friction to identify the type, as problems often provide ${\mu }_s$ for static and ${\mu }_k$ for kinetic friction.