What is the work done by a force F which acts on an object as it moves through a distance d?
W = F*d cos(θ)
Where θ is the angle between the force and the direction of motion.
Since work is proportional to cos(θ), only the component of force parallel to the motion contributes to the work; any forces perpendicular to motion do no work since cos(90)=0.
What are the units of work, as defined in physics?
Work has units of Joules (J).
Remember: work is defined as F * d cos(θ), which has units of N * m, or kg * m2/s2, which is identical to the units of Joules.
What is the sign of work done on an object, if it begins at rest and an applied force accelerates it to a speed v?
The work done on the object by the force is positive (+).
By definition, work done by a force that leads to a change in distance (hence an increase in speed) is positive work.
What is the sign of work done on an object, if it begins at speed v and an applied force decelerates it to rest?
The work done on the object by the force is negative (-).
By definition, work done by a force (such as friction) that leads to a decrease in speed is negative work.
What kind of work, positive or negative, can a person do by pushing on a box?
A person can do either positive or negative work by pushing on a box.
If the person pushes on the box in the same direction that it moves, accelerating it, then positive work is done. If the person pushes in the opposite direction of the moving box, decelerating it, then negative work is done.
What kind of work (positive, negative, or both) can frictional forces do?
Frictional forces can only do negative work.
By definition, frictional forces are always in the opposite direction of an object's motion. Hence, they can only slow the object down, and only do negative work.
What is the work done by gravity as an object of mass m moves from the ground to a height h?
W = mgh
Remember, W = F * d cos(θ). In this case, as an object moves straight up, θ = 0º and cos(θ) = 1. So, W = F * d. The force of gravity is simply mg, and the distance is h, so the total work done is mgh.
An object at rest is moved from the ground to a height h/2, then to rest at a height h. How much work does gravity do during this process?
W = -mgh
The work done by gravity while the object moves to h/2 is -mg(h/2). The work done while the object is moving from h/2 to h is also -mg(h/2). Work is negative in this case, because gravity works against the object being moved.
Notice that this is identical to the work done if the object is moved directly to h; the work done by gravity is path-independent.
Mechanical advantage is the multiplication of a force using a mechanical device. A small force exerted over a large distance is transformed into a larger force over a smaller distance.
On the AP exam, mechanical advantage appears primarily in problems including levers.
What is the relationship between force and distance in any system which includes mechanical advantage?
F1d1 = F2d2
In any system exhibiting mechanical advantage, the force exerted and the distance covered are inversely proportional,
A force F1 is exerted a distance d1 from the fulcrum of a lever. What does the force F2 at d2 from the other end equal?
F2 = F1d1/d2
In any system exhibiting mechanical advantage, the force exerted and the distance are inversely proportional, F1d1 = F2d2. Rearranging yields the above relationship.
A pulley system is set up that allows a weight m to be lifted using a force of only (1/3)mg. How far must the string on which the force is exerted be pulled in order to move the weight a distance d?
The string must be pulled a distance 3d.
In any system exhibiting mechanical advantage, the force exerted and the distance are inversely proportional, F1d1 = F2d2. So if one-third of the force is required, the distance must increase by the same proportion.
power, as used in physics
Power is a measure of the rate of energy flow.
Power is defined as energy divided by time:
P = E/t
The units of power are Watts, where 1 W = 1 J/s.
What is the power flowing through a wire, if 1,000 J of energy flow through in 0.1 s.
P = 10,000 W = 10 kW
Power is energy divided by time;
1000 J / 0.1 s = 10,000 W
Note that kW is a commonly-used unit on the AP exam.
the kinetic energy of an object
An object's kinetic energy is the energy resulting from its motion.
Kinetic energy is defined as:
KE = ½mv2
where m is the object's mass and v is its speed. The units of kinetic energy are Joules, just like all other forms of energy.
If objects 1 and 2 are moving at the same speed, but object 2 has twice the mass of object 1, how do their kinetic energies compare?
KE2 = 2*KE1
Kinetic energy is defined as ½mv2, so it is directly proportional to the mass. If the objects' speeds are the same, kinetic energy will increase proportionally with the mass.
If objects 1 and 2 are the same mass, but object 2 is moving at twice the speed of object 1, how do their kinetic energies compare?
KE2 = 4*KE1
Kinetic energy is defined as ½mv2, so it is directly proportional to the square of the velocity. If the objects' masses are the same, kinetic energy will increase with the square of the velocity.
the work-energy theorem
The work-energy theorem states that the net work done on an object by all the forces acting on it equals the change in the object's kinetic energy.
Wnet = ΔKE
An object is at rest, and a person pushes on it, doing 50 J of work that convert to motion. What is the object's final kinetic energy? (assume no dissipative forces)
The object's final KE is 50 J.
According to the work-energy theorem, the net work done on the object results in a change in its kinetic energy. Since the person is exerting the only force on the object which causes motion, the work done equals the change in kinetic energy.
An object is moving with 100 J of kinetic energy, and a frictional force acts on it, doing 50 J of work. What is the object's final kinetic energy?
The object's final KE is 50 J.
According to the work-energy theorem, the net work done on the object results in a change in its kinetic energy. Since the frictional force is the only force affecting motion, the work done equals the change in kinetic energy. Frictional forces can only do negative work, so the final kinetic energy is lowered.
An object with mass m falls through a distance h. What is its final kinetic energy? (assume no dissipative forces)
The object's final KE is mgh.
According to the work-energy theorem, the net work done on the object results in a change in its kinetic energy. Since gravity is the only force acting on the object, the change in kinetic energy simply equals the work done. Work = F * d, or, in this case, mg * h. You might also recognize this as the gravitational potential energy of the object before it fell.
the gravitational potential energy of an object in a gravitational field
An object's gravitational potential energy Ugr is defined as the work that was needed to raise it to its present height above an arbitrary reference point.
On the AP exam, the reference point will always be ground level, unless otherwise stated.
What is its gravitational potential energy of an object of mass m that is a height h above the surface of the Earth?
Ugr = mgh
This is identical to the work needed to raise the object that height, and is also equal to the kinetic energy the object will gain if it is allowed to fall freely to the ground.
How does the gravitational potential energy of a mass change, if its height above the surface of the Earth doubles?
The gravitational potential energy of the mass doubles.
Since gravitational potential energy of an object near Earth's surface Ugr = mgh is proportional to height above the ground, changing the height changes the energy by the same amount.
What is the potential energy of gravitational attraction between two objects of masses m1 and m2?
Ugr = -Gm1m2/r
G is the universal gravitational constant, and r is the distance between the centers of mass for the two objects.
This expression applies in all scenarios. Hence, is different from Ugr=mgh, which only approximates that near the Earth's surface, where gravitational acceleration is roughly constant (equal to g, ~10 m/s2).
How does the gravitational potential energy between two masses change, if the distance between the two masses doubles?
The gravitational potential energy between the masses increases.
Be careful of signs! The general expression for gravitational potential energy, Ugr = -Gm1m2/r, is always negative. Increasing r makes the fraction smaller; in this case, less negative, reflecting an increase in potential energy.
What is the force on a mass m, attached to a stretched spring, at a distance x from equilibrium?
F = -kx
Where k is the spring constant (or force constant) of the spring (in N/m) and x is the distance from equilibrium. Notice that the negative sign means force will always act in the opposite direction of net displacement.
If spring 1 has twice the value of k that spring 2 has, which spring will need to be moved the greater distance in order to create the same force?
Spring 2 (the lower k) will need to be displaced further.
Since F=-kx and spring 2 has 1/2 the k value of spring 1, spring 2 will need to move twice the distance in order to create the same value of F.
the potential energy of a spring which has been stretched
A spring's potential energy Usp is defined as the work needed to stretch or compress the spring from its equilibrium length to the current length.
On the AP exam, the spring's equilibrium length will be defined as x = 0.
What is a spring's potential energy, if a mass m is attached and the spring is stretched to a distance x from equilibrium?
Usp = ½kx2
Where x is the distance from equilibrium and k is the spring constant. There is no difference between sign for compression and expansion, since x2 will always be positive.
How does the potential energy of a spring change, if its extension from equilibrium doubles?
The potential energy of the spring increases by a factor of 4.
Since the potential energy of the spring Usp = ½kx2 is proportional to the square of the displacement of the spring, stretching the spring increases the potential energy by the square of the proportional length change.
the total mechanical energy of a physical system
The total mechanical energy of a physical system is the sum of the kinetic and potential energies of all the objects which make up the system.
Etotal = KEtotal + PEtotal
On the AP exam, you will rarely have to track more than one kinetic and one potential energy in any system.
What specific energies constitute the total mechanical energy of a cannonball in flight?
The kinetic energy from the cannonball's velocity, and the gravitational potential energy due to its height above the ground.
Note: assume the canonball is not rotating, unless explicitly given. Rotational kinetic energy is rarely tested on the AP B exam (though it's possible on the C exam).
What specific energies constitute the total mechanical energy of a mass oscillating on a horizontal spring?
The kinetic energy from the mass's velocity, and the potential energy due to the extension or compression of the spring from equilibrium.
the principle of conservation of energy
The principle of conservation of energy states that, in the absence of dissipative forces, a system's total mechanical energy is constant.
Dissipative forces such as friction or air resistance will decrease a system's total mechanical energy.
At the bottom of the initial hill, a roller coaster car is moving with speed v. Ignoring friction, what determines whether it can glide to the top of the next hill?
The car will reach the top of the next hill if its kinetic energy at the bottom Ki is greater than the gravitational potential energy will be at the top of the hill Uf.
Energy conservation states that total energy K+Ugr remains constant. At the bottom of the hill, potential energy is zero. If Ki > Uf, then the initial kinetic energy is sufficient to overcome the force of gravity and climb the hill. If Ki < Uf, the car would not make it to the top of the next hill.
a conservative force in physics
A conservative force is one which does not dissipate mechanical energy.
When only conservative forces are acting on a system, the system's total mechanical energy will stay constant.
What are some common examples of conservative forces in physics?
The most common conservative forces on the AP exam are gravity, the electrostatic and magnetic forces, and the restorative force of a stretched spring.
A spring with a mass on each end is floating in space, and one mass is pushed towards the other to compress the spring and then released. How does the total mechanical energy of the compressed spring compare to the oscillating spring?
The total mechanical energy will be the same.
The compressed spring has only potential energy of position, and the oscillating spring can only have as much kinetic energy at equilibrium as was already present. Since no dissipative forces exist, and springs are conservative, total mechanical energy will remain constant.
a nonconservative force in physics
A nonconservative force is one which dissipates mechanical energy.
When nonconservative forces are acting on a system, the system's total mechanical energy will decrease.
What are some common examples of nonconservative forces in physics?
The most common nonconservative forces on the AP exam are friction, air resistance, and fluid viscosity.
A box is pushed at some speed across the floor by a force and then released. How does the total mechanical energy of the box at the point of release compare to the box at a later time?
The total mechanical energy later will be less than when released.
The box only has kinetic energy due to velocity, but the dissipative force of friction acts between the box and the floor. Friction inhibits motion, thus kinetic energy, and total mechanical energy, will be less the longer friction acts.
The net force that an object is experiencing, is its current acceleration times its mass. Force can also be thought of as the change in momentum per unit time, or mass times the change in velocity per unit time.
The SI unit of force is the Newton (N),
1N = 1 kg*m/s2
What is the force on an object if it has:
- m = 2kg, a = 10 m/s2?
- m = 4kg, ∆v = 20 m/s, ∆t = 1s?
∆p = 10kg*m/s, ∆t = 10 m/s?
20 N (F = ma = 2 (10) = 20)
80 N (m∆v/∆t = 4 (20/1) = 80)
1 N (∆p/∆t = 10 / 10 = 1)
Note: force can be positive or negative, depending on direction of velocity.
The momentum, p, of an object is its mass times its current velocity. Momentum can also be thought of as the inertia of an object, or the mass times change in distance per unit time.
The SI unit of momentum is kg*m/s
What is the momentum of an object if it has:
- m = 2kg, v = 15 m/s?
- m = 4kg, v = 20 m/s?
m = 10kg, v = 0 m/s?
30 kg*m/s (mv = 2 (15) = 30)
80 kg*m/s (mv = 4 (20) = 80)
0 kg*m/s (mv = 10 (0) = 0)
Note: momentum can be positive or negative, depending on direction of velocity.
Impulse is the change in momentum of an object, due to a net force applied over some amount in time.
The SI unit of impulse is N*s, or kg*m/s.
What is the impulse on an object if it has:
- Fnet = 2 N, Δt = 2s?
- Δmv = 8 kg*m/s, Fnet = 4 N?
m = 10, a = 2 m/s2, Δt = 2 s?
4 kg*m/s (F*Δt = 2 (2) = 4)
8 kg*m/s (Δmv = 8 ... given)
40 kg*m/s (ma*Δt = 10 (2*2) = 40)
Note: impulse can be positive or negative, depending on direction of velocity change.
What corresponding change in momentum will cause an object to suddenly experience half as much force on it?
Momentum per unit time must be halved.
Since force = ma = mΔv/t = Δp/t, momentum and force are directly proportional. Halving force requires momentum to be halved.
What is the momentum change for an object which experiences a force that doubles its velocity?
Momentum will be doubled.
Since Δp = mΔv, a change in momentum is directly proportional to a change in velocity. Doubling one will double the other.
Describe the difference in impulses if a mass experiences a force for a given time, vs half that force for twice the length of time.
The impulses are equal.
Original impulse = F1*t1
Since F2 = F1/2 and t2 = 2t1,
then: new impulse = F2*t2 =
= (F1/2)*(2t1) = F1*t1 .
Describe why airbags reduce the force on passengers in a car accident.
Airbags lengthen the amount of time that a crash victim experiences the deceleration forces, meaning the forces are decreased.
From Impulse = Fnet*Δt, force and time are inversely proportional. Therefore, if the time increases, the force is decreased.
Describe Newton's first law of motion with regards to force.
In order for an object to have its motion changed there must be an net force acting on it.
If all forces cancel to zero, the object will not experience any change in motion.
Describe Newton's first law of motion with regards to momentum.
An object with directional momentum will continue with that momentum unless acted on by a net force.
Similarly, an object with zero momentum will continue to remain at rest until acted on by a net force.
Describe, in terms of force, what will cause an object at rest to gain some velocity?
The object must have experienced a net force.
Since all forces cancel to be zero at rest, there must have been a net force to change velocity.
Describe, in terms of momentum, what will cause an object at rest to gain some velocity?
The object must have experienced an impulse (change in momentum) from a net force.
Since all forces cancel to be zero at rest, there must have been a net force to change velocity. If there is a change in velocity, that is directly proportional to a change in momentum.
translation (as it applies to physics)
Translation is the net movement or change in location of an object. This may be thought of as a non-zero displacement in a direction.
Ex: a hockey puck may be said to have experienced translation when it began at one end of an ice rink and was later at the other end.
Translational equilibrium occurs when the sum of all of the forces acting on an object cancel to zero. The object will not experience any force causing it to translate.
The trivial case for this is an object at rest, but a object moving with constant velocity is also at translational equilibrium.
A skydiver is falling from a plane and has yet to reach terminal velocity. Why is the skydiver NOT in translational equilibrium?
The skydiver is still accelerating (changing velocity), which must be due to a non-zero net force. There is no translational equilibrium.
The downward force due to gravity is greater than the upward force due to air resistance, resulting in a net force on the skydiver.
fulcrum and lever arm
The fulcrum is the point of an object that remains fixed during rotation.
The lever arm is the distance from the fulcrum where a force is applied, to create rotation around the fulcrum.
Torque is the ability of a force to create a change in the angular orientation of an object, by rotation about a fulcrum point. Torque can also be thought of as the component of work perpendicular to a lever arm.
The SI unit of torque is the N-m.
Describe the relationship between the lever arm radius and force applied, which leads to torque.
τ = r*F sinθ
Where r is the distance between force applied and the fulcrum point, F is the magnitude of force applied, and θ is the angle between the lever arm and the force vector.
Only the component of force perpendicular to the lever arm will generate torque, since sin0 = sin180 = 0.
How will the torque on a lever arm change, if the same force is applied half the original distance from the fulcrum?
Torque will be halved.
Since torque = r*F sinθ, torque is directly proportional to the distance along the lever arm. Halving the distance that the force is applied at will halve the torque as well.
Describe the convention of rotation that is exibited from a positive torque.
Counter-clockwise rotation is positive torque.
The maximum positive (counter-clockwise) torque that a force can deliver is at 90 degrees to the lever arm.
Describe the convention of rotation that is exibited from a negative torque.
Clockwise rotation is negative torque.
The maximum negative (clockwise) torque that a force can deliver is at 90 degrees to the lever arm.
Rotational equilibrium occurs when the net torque on an object is zero. If the sum of all torques acting on an object cancel to zero then the object will not undergo rotational acceleration.
The trivial case for this is an object at rest, but a object rotating with constant spin is also in rotational equilibrium.
A child on a see-saw leans back, causing her to to lower and her partner to rise. Is the see-saw system in rotational equilibrium?
No, the see-saw is experiencing changing rotation, which must be due to a non-zero net torque, hence there is not rotational equilibrium.
The downward force due to one child creates a torque that is not balanced by the other child, resulting in a net acceleration of the see-saw.
A child of mass 40kg sits 1.5m from the pivot of a see-saw. To balance exactly, where on the opposite end should her 30kg little sister sit? (assume the seesaw is massless)
She should sit 2m from the pivot point.
Since all forces due to gravity are perpendicular to the see-saw, torque from the 40kg child will be: (mg)r = 40(10)(1.5) = 600 Nm. To balance, the 30kg child needs to supply 600 Nm of torque, which requires a distance of 2m.
On the AP B exam, torque questions that require calculation always ask for the state of rotational equilibrium (on the AP C exam, further calculations are possible).
What visual tool can be used to account for all forces acting on an object?
A free-body diagram.
Free-body diagrams represent force vectors with arrows pointing in the corresponding direction. Larger forces are represented by longer arrows.
Describe/draw the free-body diagram for a box with mass m, accelerating towards the earth in free fall.
A force due to gravity, pointing directly downwards with magnitude mg.
A force due to air resistance, pointing directly upwards with magnitude kv2 (the negative sign is only necessary for calculations).
Note: since the mass is accelerating downwards, the arrow for gravity is represented larger than for air resistance.
Describe/draw the free body diagram for the stationary block of mass m on the slope below, providing specific values where possible.
A force due to gravity pointing directly downwards, with magnitude mg.
A normal force pointing perpendicular to the slope, with magnitude mg cos(θ).
A static friction force pointing parallel to the slope, opposing gravity, with a magnitude of mg sin(θ).
Note: the length of the arrows should be proportional to the magnitude of each force.
Apparent weightlessness occurs when there is no force opposing gravity, and the object is in free-fall.
Ex: A passenger in an airplane which is in a free-fall dive, accelerating towards the earth at 9.8 m/s2, will experience apparent weightlessness.
Real weightlessness occurs when the sum of all gravitational forces acting upon an object is zero.
This is an idealized concept that would require an object to be exactly at the center of mass for the entire universe. As soon as that object was in any other location, it could still have apparent weightlessness.
A skydiver leaps from a plane, and assuming there is no air resistance, are they experiencing weightlessness?
Yes, apparent weightlessness.
There is no force opposing gravity, and the skydiver is not in contact with any objects that inhibit their inertial movement.
the principle of conversation of linear momentum
The principle of conservation of linear momentum states that in an isolated system the initial momentum on any axis will be the same as the final momentum on that axis.
Note: this principle holds true no matter what type of interaction occurs in the system.
Explain why in an isolated box of gas, individual molecules can gain or lose momentum, but the system of gas doesn't change momentum.
Since this is an isolated system, there are no external forces that act on any molecules within it. With no way to add or remove momentum from the system, momentum must be conserved.
Note: this means that if one molecule gains momentum, another must lose exactly that much momentum.
Explain with an example whether it is possible for objects in an isolated system to collide and stop.
It is possible.
Consider two cars of equal weight and equal-but-opposite velocity. Conceptually, this means that the system's initial momentum is zero, since each car's momentum is equal and opposite the other's. If they collide and stick (totally inelastic collision), then the final object's momentum must be zero - hence, both cars come to rest together.
An elastic collision is one in which the mechanical (kinetic) energy and momentum of the system are both conserved.
Elastic collisions require objects to bounce off each other perfectly, in rigid fashion, without any deformation.
Two ideal hockey pucks of equal mass, one traveling to the right and the other to the left, collide in space. Describe the motion of the pucks after collision, assuming they began with equal speed.
The pucks will bounce off of each other in a perfectly elastic collision.
The puck originally traveling left will now go back right, and the puck originally traveling right will now go back left. Their speeds and masses remain constant, but the direction (sign) of velocity and momentum will now be opposite.
An inelastic collision involves some mechanical (kinetic) energy of the system being converted to internal energy, as the objects collide and separate. Momentum will remain conserved.
An inelastic collision results from friction causing heat, or compression causing potential energy or work on an object. At least one object will be deformed, heated, broken, or otherwise changed.
Two cars of equal mass, one traveling to the right and the other to the left, collide and deform in space. Describe the motion of the cars after collision, assuming they began with equal speed and are not stuck together afterwards.
The cars will bounce off of each other in an inelastic collision.
The cars will lose kinetic energy due to the collision. Their speeds will decrease and masses may change if one car loses mass to the other. The direction (sign) of velocity and momentum will now be opposite, but total momentum remains conserved.
totally inelastic collision
A totally inelastic collision involves some mechanical (kinetic) energy of the system being converted to internal energy, as the objects stick and remain connected. Momentum will remain conserved.
An inelastic collision results in at least one object being deformed, heated, broken, or otherwise changed.
Two soft clay balls of equal mass, one traveling to the right and the other to the left, collide and stick in space. Describe the motion of the balls after collision, assuming they began with equal speed.
The balls will stick together and stop, in a totally inelastic collision.
The balls will lose kinetic energy due to the collision. Their speeds will decrease to zero and masses combine together. The velocity will now be zero, but total momentum remains conserved (it began as zero).
Two balls of equal mass collide in space. Knowing nothing else, what can be said about this system after collision?
What if the collision was elastic vs. totally inelastic?
Momentum will be conserved, no matter what collision type.
If the collision is elastic, kinetic energy will also be conserved.
If the collision is totally inelastic, the system will have one final object with double the mass.
What is the equation relating momentum before and after a totally inelastic collision, for two masses traveling at some velocities?
m1v1 + m2v2 = (m1+m2)vf
This is simply derived from the usual conservation of momentum, keeping in mind that after a totally inelastic collision the objects combine into a single mass.
What is the equation relating kinetic energies before and after a totally inelastic collision, for two masses traveling at some velocities?
½m1v12 + ½m2v22 > ½(m1+m2)vf2
This is simply derived from the usual decrease in kinetic energy in any inelastic collision, keeping in mind that after a totally inelastic collision the objects combine into a single mass.