Friction, Tension, & Gravity

General Problem Solving Approach

• When an answer doesn't match a listed option, show your work and state what you got.

• For problems involving motion with varying speeds, you can apply constant acceleration formulas in components. For example, if a path is divided into three parts: the first third with increasing speed, the second third with constant speed, and the last third with decreasing speed, you can apply acceleration equals constant formulas three times for each segment.

Types of Forces: Friction

• There are three types of friction:

  • Static Friction (f_s): Occurs when an object is at rest but a force is applied attempting to move it. It opposes the applied force and prevents motion.

  • Kinetic Friction (f_k): Occurs when an object is in motion, sliding over a surface. It always opposes the direction of motion.

  • Rolling Friction: Occurs when an object rolls over a surface (not discussed in detail in this segment).

Kinetic Friction (f_k)

• Explanation: When an object is moving (sliding), kinetic friction acts opposite to its velocity.

• Independence from Velocity: Kinetic friction does not depend on the object's velocity. Whether an object is moving at 10 ext{ mph} or 15 ext{ mph}, the kinetic friction value remains the same.

• Dependency: It primarily depends on the normal force (N) the object experiences from the surface.

• Formula: fk = \muk N

  • \mu_k (mu sub k) is the coefficient of kinetic friction, a constant that depends on the materials of the two contacting surfaces. It is typically found in tables.

  • N is the normal force, which is the force exerted by a surface perpendicular to the object resting on it.

• Example (Flat Surface):

  • For a box moving on a flat horizontal surface, the normal force (N) is equal to the gravitational force (mg) because there is no vertical acceleration. Thus, N = mg.

  • Therefore, fk = \muk mg. The friction is proportional to the object's mass and the acceleration due to gravity.

• Example (Object on a Ramp):

  • When an object slides down a ramp inclined at an angle \theta, the forces acting are: gravitational force (mg) vertically downwards, normal force (N) perpendicular to the ramp surface, and kinetic friction (f_k) opposing motion along the ramp, pointing upwards.

  • Choosing Axes: It's convenient to choose a coordinate system where the x-axis is parallel to the ramp (e.g., pointing downwards along the ramp for motion analysis) and the y-axis is perpendicular to the ramp.

  • Forces along Y-axis: The normal force (N) acts along the positive y-axis. The gravitational force (mg) has a component perpendicular to the ramp, which is -mg \cos\theta (pointing into the ramp, along negative y-axis).

  • If there's no acceleration perpendicular to the ramp (a_y = 0), then N - mg \cos\theta = 0 \implies N = mg \cos\theta.

  • Important Note: In this case, N is not equal to mg. The normal force depends on the angle of inclination.

  • Forces along X-axis: The kinetic friction (f_k) acts along the negative x-axis (up the ramp). The gravitational force (mg) has a component parallel to the ramp, -mg \sin\theta (pointing down the ramp if x-axis is upwards, or mg \sin\theta if x-axis is downwards). Let's assume the x-axis points down the ramp for an object sliding down.

  • The net force along the x-axis is F{net,x} = mg \sin\theta - fk. According to Newton's Second Law, F{net,x} = max.

  • Substituting fk = \muk N and N = mg \cos\theta: mg \sin\theta - \muk mg \cos\theta = max.

  • Special Case (Constant Velocity): If the object slides down the ramp at a constant velocity, then its acceleration (a_x) is zero.

    • mg \sin\theta - \mu_k mg \cos\theta = 0

    • mg \sin\theta = \mu_k mg \cos\theta

    • \mu_k = \frac{\sin\theta}{\cos\theta} = \tan\theta

    • This means if you know the coefficient of kinetic friction (\muk) for two surfaces, you can find the angle (\theta = \arctan(\muk)) at which an object will slide down that ramp at a constant speed.

Static Friction (f_s)

• Explanation: Static friction prevents an object from moving when an external force is applied. It acts in the opposite direction of the attempted motion.

• Adjustable Nature: Unlike kinetic friction, static friction is not a constant value. It adjusts itself to be equal and opposite to the applied force, up to a certain maximum value.

  • If you push a heavy box with 10 \text{ N} and it doesn't move, the static friction is 10 \text{ N}.

  • If you push with 50 \text{ N} and it still doesn't move, the static friction is 50 \text{ N}.

• Maximum Static Friction (f_{s,max}): There is a limit to how much static friction can oppose an applied force. Once the applied force exceeds this maximum, the object begins to move, and kinetic friction takes over.

• Formula for Maximum Static Friction: f{s,max} = \mus N

  • \mus (mu sub s) is the coefficient of static friction, which is also a constant depending on the surfaces. Generally, \mus \ge \mu_k.

  • Therefore, static friction always satisfies the inequality: fs \le \mus N.

Static vs. Kinetic Friction (Graphical Representation)

• Imagine plotting the friction force against the applied push force:

  • Static Zone: As the push force increases from zero, the static friction force increases linearly, exactly matching the push force. This continues until the push force reaches the maximum static friction (f_{s,max}).

  • Transition: At f_{s,max}, the object starts to move. This is often followed by a slight dip in the friction force.

  • Kinetic Zone: Once the object is moving, the friction force becomes kinetic friction (fk). Since fk is largely independent of the applied push (only depends on N), it remains relatively constant, represented by a flat line lower than f_{s,max}. This dip explains why it's harder to get an object moving than to keep it moving.

Coefficients of Friction (Examples)