Physics: Kinematics and Dynamics (Vectors/Scalars, Newton's Laws, and Forces)
Match the following: A. Newton's First Law B. Newton's Second Law C. Newton's Third Law - An object of mass m will accelerate when the vector sum of the forces results in some nonzero resultant force vector. No acceleration will occur when the vector sum of the forces results in a cancellation of those forces. Note that the net force and acceleration vectors necessarily point in the same direction. - To every action, there is always an opposite but equal reaction. This law is also known as the law of action and reaction. - A body at rest or in motion with constant velocity will remain that way unless acted upon by a net force. This law is also known as the law of inertia. · FAB = -FAB · Fnet = ma · Fnet = ma = 0
A. Newton's First Law - A body at rest or in motion with constant velocity will remain that way unless acted upon by a net force. This law is also known as the law of inertia. · Fnet = ma = 0 B. Newton's Second Law - An object of mass m will accelerate when the vector sum of the forces results in some nonzero resultant force vector. No acceleration will occur when the vector sum of the forces results in a cancellation of those forces. Note that the net force and acceleration vectors necessarily point in the same direction. In other words, acceleration results from the sum of the force vectors. · Fnet = ma C. Newton's Third Law - To every action, there is always an opposite but equal reaction. This law is also known as the law of action and reaction. · FAB = -FAB
What is the difference between average velocity and average speed?
Average velocity is a vector quantity and thus has magnitude and direction. Average velocity is the ratio of the displacement vector over the change in time. Average speed is a scalar quantity and thus only has magnitude and no direction. Average speed is the ratio of the total distance traveled over the change in time. Average speed accounts for actual distance traveled, whereas average velocity does not.
Which of the following is not considered a vector quantity? A. Acceleration B. Energy C. Force D. Displacement
B. Energy Vectors are numbers that have magnitude and direction. Vector quantities include: - Displacement - Velocity - Acceleration - Force Scalars are numbers that have magnitude only and no direction. Scalar quantities include: - Distance - Speed - Energy - Pressure - Mass
Which type of friction does a wheel rolling along a road experience? A. Kinetic Friction B. Static Friction C. Both A and B D. None of the above
B. Static Friction A wheel rolling along a road does not experience kinetic friction because the tire is not actually sliding against the pavement. The tire maintains an instantaneous point of static contact with the road and, therefore, experiences static friction. Only when the tire begins to slide on, say, an icy patch will kinetic friction come into play.
Displacement and Velocity
Displacement and Velocity
What is the difference between distance and displacement?
Distance accounts for the entire journey; whereas, displacement only accounts for the difference between the initial (start) point and the end point. For example, earth travels roughly 940 million kilometers over the course of one year. However, because this is a circular path, the displacement of the Earth in one year is zero kilometers. Understand that displacement does not account for the actual pathway taken between the initial and the final positions - only the net change in position from initial to final. Displacement is a vector quantity. Distance traveled, on the other hand, considers the pathway taken and is a scalar quantity.
Equation of Average Velocity average velocity = Δx / Δt Δx = change in position Δt = change in time Velocity is a vector quantity and is measured in SI units of meters per second (m/s).
Equation of Average Velocity average velocity = Δx / Δt Δx = change in position Δt = change in time Velocity is a vector quantity and is measured in SI units of meters per second (m/s).
True or False: In uniform circular motion, the tangential force is zero.
True. In uniform circular motion, the tangential force is zero because there is no change in the speed of the object.
True or False: The direction of the velocity vector is necessarily the same as the direction of the displacement vector.
True. The direction of the velocity vector is necessarily the same as the direction of the displacement vector.
True or False: The instantaneous speed of an object will always be equal to the magnitude of the objects instantaneous velocity.
True. The instantaneous speed of an object will always be equal to the magnitude of the objects instantaneous velocity, which is a measure of the average velocity as the change in time approaches zero.
True or False: The sum of two or more vectors is called the resultant of the vector.
True. The sum of two or more vectors is called the resultant of the vector. One way to find the sum or resultant of two vectors A and B is to place the tail of B at the tip of A without changing either the length or the direction of either arrow. In this tip-to-tail method, the lengths of the arrows must be proportional to the magnitudes of the vectors. The vector sum A + B is the vector joining the tail of A to the tip of B and pointing toward the tip of B.
True or False: The value of μs is always larger than the value of μk.
True. The value of μs is always larger than the value of μk. Therefore, the maximum value for static friction will always be greater than the constant value for kinetic friction: objects will "stick" until they start moving, and then will slide more easily over one another.
Vectors and Scalars
Vectors and Scalars
Equation/Math Tip: Inclined Planes For inclined plane questions, divide force vectors into components that are parallel and perpendicular to the plane: Fgparallel = mgsinθ Fgperpendicular = mgcosθ where Fgparallel is the component of gravity parallel to the plane (oriented down the plane) ---> use this to calculate acceleration Fgperpendicular is the component of gravity perpendicular to the plane (oriented into the plane) ---> use this to calculate normal force.
Equation/Math Tip: Inclined Planes For inclined plane questions, divide force vectors into components that are parallel and perpendicular to the plane: Fgparallel = mgsinθ ---> use to calculate acceleration along with Fnet = ma Fgperpendicular = mgcosθ ---> use to calculate normal force where Fgparallel is the component of gravity parallel to the plane (oriented down the plane) ---> use this to calculate acceleration Fgperpendicular is the component of gravity perpendicular to the plane (oriented into the plane) ---> use this to calculate normal force.
Equation: Average Acceleration average acceleration = Δv/Δt Δv = change in velocity Δt = change in time Acceleration is the rate of change of velocity that an object experiences as a result of some applied force. Acceleration, like velocity, is a vector quantity and is measured in SI units of meters per second squared (m/s²). On a graph of velocity vs. time, the tangent to the graph at any time t, which corresponds to the slope of the graph at that time, indicates the instantaneous acceleration (average acceleration as Δt approaches zero).
Equation: Average Acceleration average acceleration = Δv/Δt Δv = change in velocity Δt = change in time Acceleration is the rate of change of velocity that an object experiences as a result of some applied force. Acceleration, like velocity, is a vector quantity and is measured in SI units of meters per second squared (m/s²). On a graph of velocity vs. time, the tangent to the graph at any time t, which corresponds to the slope of the graph at that time, indicates the instantaneous acceleration (average acceleration as Δt approaches zero).
Equation: Centripetal Force Fc = (mv²)/r Fc = magnitude of the centripetal force m = mass v = speed r = radius of the circular path As a force, the centripetal force generates centripetal acceleration. Remember that both force and acceleration are vectors and the acceleration is always in the same direction as the net force. Thus, it is this acceleration generated by the centripetal force that keeps an object in its circular pathway. When the centripetal force is no longer acting on the object, it will simply exit the circular pathway and assume a path tangential to the circle at that point.
Equation: Centripetal Force Fc = (mv²)/r Fc = magnitude of the centripetal force m = mass v = speed r = radius of the circular path As a force, the centripetal force generates centripetal acceleration. Remember that both force and acceleration are vectors and the acceleration is always in the same direction as the net force. Thus, it is this acceleration generated by the centripetal force that keeps an object in its circular pathway. When the centripetal force is no longer acting on the object, it will simply exit the circular pathway and assume a path tangential to the circle at that point.
Equation: Gravitational Force Fg = (Gm1m2)/r² G = universal gravitational constant (6.67x10^-11 Nm²/kg²) m1 and m2 = the masses of the two objects r = the distance between the two objects center of mass The magnitude of the gravitational force is inversely related to the square of the distance (that is if r is halved, then Fg will quadruple). The magnitude of the gravitational force is also directly related to the masses of the objects (that is if m1 is tripled, then Fg will triple).
Equation: Gravitational Force Fg = (Gm1m2)/r² G = universal gravitational constant (6.67x10^-11 Nm²/kg²) m1 and m2 = the masses of the two objects r = the distance between the two objects center of mass The magnitude of the gravitational force is inversely related to the square of the distance (that is if r is halved, then Fg will quadruple). The magnitude of the gravitational force is also directly related to the masses of the objects (that is if m1 is tripled, then Fg will triple).
Equation: Splitting vectors into components If θ is the angle between V (the hypotenuse) and the x-component, then cos θ = X/V and sin θ = Y/V. In other words: X = V cos θ Y = V sin θ Look at pg 7 in the physics book for diagram of the triangle if needed.
Equation: Splitting vectors into components If θ is the angle between V (the hypotenuse) and the x-component, then cos θ = X/V and sin θ = Y/V. In other words: X = V cos θ Y = V sin θ Look at pg 7 in the physics book for diagram of the triangle if needed.
Equation: Static and Kinetic Force fs ≤ μsN fs = static force μs = coefficient of static friction N = normal force ≤ indicates that there is a range of possible values for static friction (as you push an object you have to keep increasing the force of the push until the object finally moves; as you increase the force to push, static friction also increases until it can no longer keep up). fk = μkN fk = kinetic force μk = coefficient of kinetic friction N = normal force Kinetic friction exists between a sliding object and the surface over which the object slides.
Equation: Static and Kinetic Force fs ≤ μsN fs = static force μs = coefficient of static friction N = normal force ≤ indicates that there is a range of possible values for static friction (as you push an object you have to keep increasing the force of the push until the object finally moves; as you increase the force to push, static friction also increases until it can no longer keep up). fk = μkN fk = kinetic force μk = coefficient of kinetic friction N = normal force Kinetic friction exists between a sliding object and the surface over which the object slides.
Equation: The Kinematic Equations v = vo + at v² = vo² + 2aΔx Δx = vot + (at²)/2 Δx = (average velocity)t = (v + vo)/2 x t where x, v, and a are the displacement, velocity, and acceleration vectors, respectively; vo is the initial velocity; and t is time.
Equation: The Kinematic Equations v = vo + at v² = vo² + 2aΔx Δx = vot + (at²)/2 Δx = (average velocity)t = (v + vo)/2 x t where x, v, and a are the displacement, velocity, and acceleration vectors, respectively; vo is the initial velocity; and t is time.
Equation: Torque T = r x F = rF sin θ r = length of lever arm F = magnitude of the force θ = the angle between the lever arm and the force vectors Torques that generate clockwise rotation are considered negative, while torques that generate counterclockwise rotation are positive. Rotational motion occurs when forces are applied against an object in such as way as to cause the object to rotate around a fixed pivot point, also known as the fulcrum. Application of force at some distance from the fulcrum generates torque (t) or the moment of force. The distance between the applied force and the fulcrum is termed the lever arm. It is the torque that generates rotational motion, not the mere application of the force itself. Remember that sin 90° = 1. This means that torque is greatest when the force applied is 90 degrees (perpendicular) to the lever arm. Knowing that sin 0° = 0 tells us that there is no torque when the force applied is parallel to the lever arm.
Equation: Torque T = r x F = rF sin θ r = length of lever arm F = magnitude of the force θ = the angle between the lever arm and the force vectors Torques that generate clockwise rotation are considered negative, while torques that generate counterclockwise rotation are positive. Rotational motion occurs when forces are applied against an object in such as way as to cause the object to rotate around a fixed pivot point, also known as the fulcrum. Application of force at some distance from the fulcrum generates torque (t) or the moment of force. The distance between the applied force and the fulcrum is termed the lever arm. It is the torque that generates rotational motion, not the mere application of the force itself. Remember that sin 90° = 1. This means that torque is greatest when the force applied is 90 degrees (perpendicular) to the lever arm. Knowing that sin 0° = 0 tells us that there is no torque when the force applied is parallel to the lever arm.
Equation: Weight of an object Fg = mg Fg = weight of an object m = mass of the object g = acceleration due to gravity, 9.8m/s² (usually rounded up to 10m/s²)
Equation: Weight of an object Fg = mg Fg = weight of an object m = mass of the object g = acceleration due to gravity, 9.8m/s² (usually rounded up to 10m/s²)
True or False: The average speed of an object will always be equal to the magnitude of the objects average velocity.
False The average speed of an object will not necessarily always be equal to the magnitude of the objects average velocity. This is because average velocity is the ratio of the displacement vector over the change in time (and is a vector), whereas average speed (which is a scalar) is the ratio of the total distance traveled over the change in time. Average speed accounts for actual distance traveled, whereas average velocity does not.
True or False: An object moving at constant acceleration has zero velocity.
False. An object moving at constant velocity has zero acceleration. Constant velocity is also called terminal velocity.
True or False: Force can only exist between objects that touch.
False. Force can exist between objects that are not even touching, such as gravity or electrostatic forces between point charges.
True or False: Mass and weight are the same.
False. Mass and weight are not the same. Mass (m) is a measure of a body's inertia - the amount of matter in the object. Mass is a scalar quantity so only has magnitude. It has the SI unit of kilogram. Weight (Fg) is a measure of gravitational force on an object's mass. Because weight is a force, it is a vector quantity with the units of newtons (N).
True or False: The first condition of equilibrium states that rotational equilibrium exists only when the vector sum of all the torques acting on an object is zero.
False. The second condition of equilibrium states that rotational equilibrium exists only when the vector sum of all the torques acting on an object is zero. The first condition of equilibrium states that translational equilibrium exists only when the vector sum of all of the forces acting on an object is zero, and it is merely a reiteration of Newton's first law.
True or False: Scalars are numbers that have magnitude and direction.
False. Vectors are numbers that have magnitude and direction. Scalars are numbers that have magnitude only and no direction.
Forces and Acceleration
Forces and Acceleration
Math Tip: Projectile Motion For projectile motion, when given the angle of elevation and the initial velocity, you have to consider the x and y components: For the y component: to get the initial velocity of the y component calculate: voy = vo sin degrees elevated For the x component: to get the initial velocity of the x component calculate: vox = vo cos degrees elevated For example, if a projectile is fired from ground level with an intial velocity of 50 m/s and and initial angle of 37°. Assuming g = 10 m/s² (Note: sin 37° = 0.6; cos 37° = 0.8). voy = vosin37° = (50m/s)(0.6) = 30 m/s *now you can plug in to any kinematic equation to solve for other things: y = voyt + (ayt²)/2 0 = (30m/s)(t) + (-10m/s²)(t)²/2 5t² = 30t t=0 or 6s The height of the ball is zero at 0 seconds (its initial position) and 6 seconds (when it hits the ground) For the x component: vox = vocos 37° = (50m/s)(0.8) = 40 m/s *now we can plug in: x = vxt + (axt²)/2 = (40m/s)(6s) + 0 = 240m = total distance traveled -its + 0 because the horizontal acceleration is 0; a projectile only has a vertical acceleration due to gravity.
Math Tip: Projectile Motion For projectile motion, when given the angle of elevation and the initial velocity, you have to consider the x and y components: For the y component: to get the initial velocity of the y component calculate: voy = vo sin degrees elevated For the x component: to get the initial velocity of the x component calculate: vox = vo cos degrees elevated For example, if a projectile is fired from ground level with an intial velocity of 50 m/s and and initial angle of 37°. Assuming g = 10 m/s² (Note: sin 37° = 0.6; cos 37° = 0.8). voy = vosin37° = (50m/s)(0.6) = 30 m/s *now you can plug in to any kinematic equation to solve for other things: y = voyt + (ayt²)/2 0 = (30m/s)(t) + (-10m/s²)(t)²/2 5t² = 30t t=0 or 6s The height of the ball is zero at 0 seconds (its initial position) and 6 seconds (when it hits the ground) For the x component: vox = vocos 37° = (50m/s)(0.8) = 40 m/s *now we can plug in: x = vxt + (axt²)/2 = (40m/s)(6s) + 0 = 240m = total distance traveled -its + 0 because the horizontal acceleration is 0; a projectile only has a vertical acceleration due to gravity.
Mechanical Equilibrium: Dynamics The study of forces and torques is called dynamics
Mechanical Equilibrium: Dynamics The study of forces and torques is called dynamics
Motion with Constant Acceleration
Motion with Constant Acceleration
Multiplying Vectors by Scalars When a vector is multiplied by a scalar, its magnitude will change. Its direction will either be parallel or antiparallel to its original direction. If a vector A is multiplied by the scalar value n, a new vector, B, is created such that B = nA. To find the magnitude to the new vector, B, simply multiply the magnitude of A by the absolute value of n. To determine the direction of the vector B, we must look at the sign of n. If n is a positive number, then B and A are in the same direction. However, if n is a negative number, then B and A point in opposite directions. For example, if vector A is multiplied by the scalar +3, then the new vector B is three times as long as A, and points in the same direction. If vector A is multiplied by the scalar -3, then B would still be three times as long as A but would now point in the opposite direction.
Multiplying Vectors by Scalars When a vector is multiplied by a scalar, its magnitude will change. Its direction will either be parallel or antiparallel to its original direction. If a vector A is multiplied by the scalar value n, a new vector, B, is created such that B = nA. To find the magnitude to the new vector, B, simply multiply the magnitude of A by the absolute value of n. To determine the direction of the vector B, we must look at the sign of n. If n is a positive number, then B and A are in the same direction. However, if n is a negative number, then B and A point in opposite directions. For example, if vector A is multiplied by the scalar +3, then the new vector B is three times as long as A, and points in the same direction. If vector A is multiplied by the scalar -3, then B would still be three times as long as A but would now point in the opposite direction.
Multiplying Vectors by Vectors to get a Scalar: The Dot Product (A·B) Two vector quantities can generate a third vector or a scalar by multiplication. To generate a scalar quantity like work, we multiply the magnitudes of the two vectors of interest (force and displacement) and the cosine of the angle between the two vectors: A · B = IAI IBI cos θ
Multiplying Vectors by Vectors to get a Scalar: The Dot Product (A·B) Two vector quantities can generate a third vector or a scalar by multiplication. To generate a scalar quantity like work, we multiply the magnitudes of the two vectors of interest (force and displacement) and the cosine of the angle between the two vectors: A · B = IAI IBI cos θ
Multiplying Vectors by Vectors to get a Vector: The Cross Product (A x B) When generating a third vector like torque, we need to determine both its magnitude and direction. To do so, we multiply the magnitudes of the two vectors of interest (force and the lever arm) and the sine of the angle between the two vectors. Once we have the magnitude we use the right-hand rule to determine its direction. A x B = IAI IBI sin θ One method of the right-hand rule: Consider a resultant C where C = A x B 1. Point your thumb in the direction of vector A 2. Extend your fingers in the direction of vector B. You may need to rotate your wrist to get the correct configuration of thumb and fingers 3. Your palm establishes the plane between the two vectors. The direction your palm points is the direction of the resultant C.
Multiplying Vectors by Vectors to get a Vector: The Cross Product (A x B) When generating a third vector like torque, wee need to determine both its magnitude and direction. To do so, we multiply the magnitudes of the two vectors of interest (force and the lever arm) and the sine of the angle between the two vectors. Once we have the magnitude we use the right-hand rule to determine its direction. A x B = IAI IBI sin θ One method of the right-hand rule: Consider a resultant C where C = A x B 1. Point your thumb in the direction of vector A 2. Extend your fingers in the direction of vector B. You may need to rotate your wrist to get the correct configuration of thumb and fingers 3. Your palm establishes the plane between the two vectors. The direction your palm points is the direction of the resultant C.
Newton's Laws
Newton's Laws
