5b 3rd midterm

When a problem asks for acceleration, what should you almost always do first?

Draw a separate free-body diagram for each object. Then write Newton's 2nd Law for each object and each useful direction.

If two objects are connected by a string over a pulley, how do you decide acceleration signs?

Pick positive directions first. The accelerations usually have the same magnitude, but signs depend on axes. For example, if (+y) is upward for the hanging mass and it moves down, then (a_y=-a).

When two blocks are connected by a string, do they always have the same acceleration?

They have the same magnitude if the string is taut and does not stretch. Components can be opposite signs depending on axes.

When two blocks are slipping relative to each other, should their accelerations be equal?

No. If they slip, they can have different accelerations. Use separate equations for each block.

For a block on top of another block that is slipping, what force usually accelerates the top block?

Kinetic friction. Usually (f_k=m_1a_1) for the top block.

If a block is sliding, what kind of friction do you use?

Kinetic friction: (f_k=\mu_kN).

If a block is not slipping but might slip, what kind of friction do you use?

Static friction. At the threshold: (f_s=f_{s,\max}=\mu_sN).

When do you set static friction equal to (\mu_sN)?

Only when the object is just about to slip or you are finding a minimum/maximum force at the slipping threshold.

Should Newton's third-law force pairs go on the same FBD?

No. They act on different objects. Equal and opposite forces belong on separate diagrams.

If an object is on a horizontal surface with no vertical acceleration, what equation should you write?

(F_{\text{net},y}=0). Use it to solve for (N), especially before finding friction.

If a force pulls upward at an angle on a block, what happens to friction?

The upward component reduces (N), so friction decreases. Use (N=mg-F\sin\theta).

If a force pushes downward at an angle on a block, what happens to friction?

The downward component increases (N), so friction increases. Use (N=mg+F\sin\theta).

Why solve for (N) before using friction?

Because (N) is not always (mg). Angled forces, inclines, or vertical acceleration can change it.

If two blocks slide relative to each other, which normal force goes into friction?

Use the normal force between the two blocks, not necessarily the floor normal.

If block 1 sits on block 2 on a horizontal surface, what is (N) on block 1?

Usually (N_{21}=m_1g), assuming no vertical acceleration and no other vertical forces.

If block 1 and block 2 are slipping, how are their friction forces related?

Same magnitude, opposite directions: (f_{k,12}=f_{k,21}).

If a problem gives (\mu_k), what does that tell you about the situation?

The objects are sliding relative to each other or against a surface.

If a problem gives (\mu_s), what should you check?

Whether the object is at rest, about to slip, or moving with another object without slipping.

What is the biggest friction mistake?

Using (f=\mu N) before knowing what (N) actually is.

If a friction force is the only horizontal force on an object, what can you immediately do?

Set (f=ma) in the horizontal direction.

For a block on an incline, what axes should you usually choose?

(+x) parallel to the incline and (+y) perpendicular to the incline.

If (+x) is up the incline, what is the sign of gravity's parallel component?

It is negative: (-mg\sin\theta), because gravity pulls down the incline.

For a block on an incline with no motion perpendicular to the surface, what is (N)?

(N=mg\cos\theta).

If a block moves up an incline, which way does kinetic friction point?

Down the incline.

If a block moves down an incline, which way does kinetic friction point?

Up the incline.

In an incline + hanging mass problem, where does (m_1+m_2) come from?

When equations are combined, both masses have acceleration terms: (m_1a+m_2a=(m_1+m_2)a).

How do you decide whether friction is positive or negative on an incline?

First decide motion/assumed acceleration direction. Friction points opposite motion, then assign sign from your axis.

If your final acceleration is negative, what does that mean?

Your assumed positive direction was wrong. The magnitude is still useful.

What should you do before writing incline equations?

Draw (mg), split it into (mg\sin\theta) and (mg\cos\theta), then place friction opposite motion.

Why is (mg\cos\theta) usually not in the acceleration equation along the incline?

It is perpendicular to the surface and is used to find (N), not the motion along the incline.

When solving connected-object problems, should you solve for tension first or eliminate it?

Usually eliminate tension if the question asks for acceleration.

When should you solve for tension directly?

After finding acceleration, or if the question asks for tension, stretch, or stress.

What does adding equations often do in connected-object problems?

It cancels internal forces like tension.

Why can tension be treated as internal for a two-block system?

The string pulls the two objects in opposite directions, so tension cancels when considering the combined system.

If a hanging mass is at rest, what is the tension?

(T=mg).

If a hanging mass accelerates downward and (+y) is upward, what equation should you use?

(T-mg=-ma), so (T=mg-ma).

If a hanging mass accelerates upward and (+y) is upward, what equation should you use?

(T-mg=ma), so (T=mg+ma).

When can you write (F=(m_1+m_2)a)?

When treating both objects as one system and (F) is the net external force in the acceleration direction.

Why can't you always use (F=(m_1+m_2)a) right away?

Other external forces may matter: friction, gravity components, angled pulls, support forces, etc.

What is the safest method for pulley/block problems?

Write one Newton's 2nd Law equation per object, connect accelerations, then solve.

If a stretched spring is between two blocks, which way does it pull each block?

It pulls each block back toward the other end of the spring: opposite directions on the two blocks.

If two identical masses connected by a spring accelerate together, how can you find spring force?

Write Newton's 2nd Law for both blocks and use that their (ma) values are equal.

If a problem asks for "distance stretched from unstressed length," what are you solving for?

The extension (s) or (\Delta L). Find spring force first, then use (s=F_{\text{sp}}/k).

Why can spring stretch be independent of mass in some two-block spring problems?

If both masses change equally, the weight terms can cancel when comparing the equations.

If a question asks "how much longer as a fraction of unstressed length," what are you finding?

Strain: (\Delta L/L_0).

In a string-stretch problem, what do you usually need before using Young's modulus?

Tension in the string.

If a string/wire has diameter (d), what area do you use?

(A=\pi(d/2)^2).

What is the order for Young's modulus problems?

Find tension → find area → find stress/strain → answer fractional stretch.

If both springs hold a bar in equilibrium, what equations do you need?

Use force balance for vertical forces and torque balance for locations/unknown forces.

If a spring's stretched length and unstressed length are given, what is the extension?

Extension = stretched length − unstressed length.

If an object moves in a circle at constant speed, does it accelerate?

Yes. It has centripetal acceleration toward the center.

In circular motion, what direction should the net force point?

Toward the center of the circle.

If a block moves in a circle on a frictionless table attached to a string, what provides centripetal force?

Tension in the string.

If the hanging mass attached to the circular-motion string is at rest, what is (T)?

(T=m_{\text{hanging}}g).

What is the strategy for a table-circle + hanging-mass problem?

Set hanging mass tension equal to centripetal force: (m_2g=m_1v^2/r).

What is the common circular-motion trap?

Thinking constant speed means zero acceleration. The direction changes, so acceleration is not zero.

If the radius increases while tension stays the same, what happens to required speed?

Larger (r) allows larger (v) for the same centripetal force.

If the speed increases, what happens to centripetal force needed?

It increases a lot because (v^2) is involved.

What should you never do in circular motion?

Do not add a fake "centripetal force" to the FBD. Centripetal force is the net inward force.

What should your FBD show for circular motion?

Real forces only, then identify which real force(s) point toward the center.

When a rigid object is in static equilibrium, what equations do you need?

(F_{\text{net},x}=0), (F_{\text{net},y}=0), and (\tau_{\text{net}}=0).

How do you choose a pivot point in torque problems?

Choose a point where unknown forces act so their torque becomes zero.

Why does a hinge force often disappear from the torque equation?

If the hinge is the pivot, its lever arm is zero.

For a uniform beam, where does its weight act?

At the center of the beam.

If a beam leans at an angle, how do you find the torque arm for weight?

Use the perpendicular distance from pivot to the weight's line of action.

If a wall is frictionless, what force can the wall exert?

Only a normal force perpendicular to the wall.

If a beam is held by a cable at an angle, what part of tension creates torque?

The component perpendicular to the beam. Usually use (rT\sin\phi).

If a horizontal beam is held by a cable making angle (\theta) with the horizontal, what torque does the cable produce?

Usually (LT\sin\theta), if attached at the end.

If (L) appears in every torque term, what may happen?

(L) may cancel. Don't assume every given length must appear in the final answer.

After finding cable tension in a beam problem, how do you find hinge force?

Use (F_{\text{net},x}=0) and (F_{\text{net},y}=0), then combine components.

If exact force direction is unknown, how should you draw it?

Break it into components, like (F_{Hx}) and (F_{Hy}).

Why do you draw an extended FBD for torque?

Because where the force acts matters, not just what forces exist.

If a force acts through the pivot, what torque does it produce?

Zero torque.

If force is parallel to the position vector (\vec r), what torque does it produce?

Zero torque.

What is the biggest torque setup mistake?

Using the full distance instead of the perpendicular lever arm.

What happens to the normal force at the support an object is about to lose contact with?

It becomes zero.

In a tipping problem, where should you place the pivot?

At the support/contact point that remains in contact.

If a rod is about to lose contact with the left support, what is true?

(N_{\text{left}}=0). The rod is about to rotate about the other support.

Why don't you need the support force at the pivot in a tipping torque equation?

It acts at the pivot, so it produces no torque.

For a uniform rod, where does the weight torque come from?

From the rod's center of mass, located at the midpoint.

What phrase should immediately make you think of tipping?

"Minimum force to lose contact" or "just begins to tip."

In tipping, why do you not set both normal forces equal to zero?

The object loses contact with one support but pivots around the other.

What is the usual tipping equation structure?

Clockwise torque = counterclockwise torque about the remaining contact point.

If a downward force is applied farther from the pivot, what happens to its torque?

Torque increases because the lever arm is larger.

What makes a tipping problem different from a regular equilibrium problem?

One support force is specifically zero at the tipping threshold.

Problem says "constant speed in a circle." What should you think?

Use centripetal acceleration. Constant speed does not mean zero acceleration.

Problem says "at rest" or "static equilibrium." What should you think?

Net force is zero, and if rotation matters, net torque is zero.

Problem says "just begins to tip." What should you think?

One support normal becomes zero. Choose pivot at the other support.

Problem says "minimum force before slipping." What should you think?

Static friction is maxed out: (f_s=\mu_sN).

Problem says "sliding." What should you think?

Use kinetic friction: (f_k=\mu_kN).

Problem says "frictionless wall." What should you think?

The wall only exerts a normal force.

Problem says "light, frictionless pulley." What should you think?

Tension is the same throughout the string.

Problem says "light string." What should you think?

The string's mass is negligible and transmits tension.

Problem says "both blocks accelerate together." What should you think?

Same acceleration; static friction may be involved.

Problem says "blocks slip relative to each other." What should you think?

Different accelerations are possible; use kinetic friction.

Problem asks "is the student correct?" about action-reaction forces. What is the likely trap?

They may be comparing third-law forces incorrectly. Third-law pairs are equal and opposite.

What should you do when a final acceleration is negative?

Interpret direction. The actual acceleration is opposite your chosen positive direction.

Best way to avoid sign mistakes?

Choose axes, label positive directions, then assign every force a sign before plugging in numbers.

When should you treat multiple objects as one system?

When internal forces cancel and you only need overall acceleration or external force.

When should you not treat multiple objects as one system?

When you need internal forces like tension, friction between objects, or contact normal.