SPH4U · Unit 1 test review

Dynamics: Test Review

Everything for the Unit 1 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Motion in Two Dimensions and Projectiles

Components: $v_{x} = v cos θ$ and $v_{y} = v sin θ$ ; recombine with $v = v_{x}^{2} + v_{y}^{2}$ and $θ = tan^{- 1} (v_{y} / v_{x})$ .
Relative velocity adds as vectors: $v_{BG} = v_{B W} + v_{W G}$ .
For a projectile with no air resistance, $a_{x} = 0$ and $a_{y} = 9.81 m/s^{2}$ downward; time is the only link between the two directions.
At maximum height $v_{y} = 0$ , but $v_{x}$ is unchanged and the acceleration is still $9.81 m/s^{2}$ down.
Level ground only: $R = \frac{v ^{2} sin 2 θ}{g}$ , maximum at 45°, equal for complementary angles.

Forces on Inclines and Friction

On an incline: $F_{g ∥} = m g sin θ$ along the slope and $F_{g ⊥} = m g cos θ$ into it.
With no other perpendicular forces, $F_{N} = m g cos θ$ .
Static friction: $F_{s} \leq μ_{s} F_{N}$ . Kinetic friction: $F_{k} = μ_{k} F_{N}$ , and usually $μ_{k} < μ_{s}$ .
Sliding down with friction: $a = g (sin θ - μ_{k} cos θ)$ .
A resting block starts to slide when $tan θ = μ_{s}$ .

Connected Objects and Frames of Reference

Treat objects moving together as a system: $a = \frac{F _{n e t, e x t er na l}}{m _{t o t a l}};$ tension is internal and cancels.
Find tension from one object's free-body diagram, for example $F_{T} = m_{2} (g - a)$ for a hanging mass accelerating down.
Atwood machine: $a = \frac{( m _{2} - m _{1} ) g}{m _{1} + m _{2}}$ .
An inertial frame is not accelerating; Newton's laws hold there using only real forces.
In an accelerating elevator the scale reads $F_{N} = m (g + a)$ with up as positive; in free fall it reads zero.

Uniform Circular Motion

Uniform circular motion has constant speed but changing velocity, so there is an acceleration toward the centre.
$a_{c} = \frac{v ^{2}}{r} = \frac{4 π ^{2} r}{T ^{2}} = 4 π^{2} r f^{2},$ with $v = \frac{2 π r}{T}$ and $f = \frac{1}{T}$ .
$F_{c} = \frac{m v ^{2}}{r}$ is supplied by real forces such as friction, tension, gravity or the normal force; it is never an extra force.
Flat curve: $v_{ma x} = μ_{s} g r$ .
Vertical circle: the minimum speed at the top is $v = g r$ ; tension is greatest at the bottom.

Banked Curves, Rotating Frames and Artificial Gravity

On a banked curve the horizontal component of the normal force, $F_{N} sin θ$ , supplies centripetal force.
Frictionless banking: $tan θ = \frac{v ^{2}}{r g}$ , independent of mass; at the design speed no friction is needed.
With friction, speeds between $v_{min}$ and $v_{ma x}$ are safe; friction points down the slope above the design speed and up the slope below it.
Use horizontal and vertical axes on a banked curve, because the acceleration is horizontal; $F_{N} = m g / cos θ$ at the design speed.
In a rotating frame, centrifugal ( $m ω^{2} r$ outward) and Coriolis (sideways on moving objects) effects are apparent, not real, forces. On Earth, Coriolis deflection is to the right in the Northern Hemisphere.
Artificial gravity: the floor of a spinning station supplies $ω^{2} r = \frac{4 π ^{2} r}{T ^{2}}$ ; for $g$ , $T = 2 π r / g$ .

Where marks get lost

Using $R = v^{2} sin 2 θ / g$ for a launch from a cliff or onto a raised target. It only works when the landing height equals the launch height. (Motion in Two Dimensions and Projectiles)
Saying the velocity is zero at the top of the path. Only the vertical component is zero; the horizontal component is unchanged. (Motion in Two Dimensions and Projectiles)
Letting the horizontal velocity slow down during the flight. With no air resistance there is no horizontal force, so $v_{x}$ is constant. (Motion in Two Dimensions and Projectiles)
Mixing a sign convention: taking up as positive for $v_{i y}$ but then using $a_{y} = + 9.81 m/s^{2}$ . (Motion in Two Dimensions and Projectiles)
Using the full boat speed as the crossing speed when the boat is angled upstream. Only the component perpendicular to the banks carries it across. (Motion in Two Dimensions and Projectiles)
Swapping sine and cosine: using $m g cos θ$ as the pull down the slope. Check with 0°: nothing should pull along a flat surface, and $sin 0^{\circ} = 0$ . (Forces on Inclines and Friction)
Taking $F_{N} = m g$ on a slope. On an incline the normal force is $m g cos θ$ , which is smaller. (Forces on Inclines and Friction)
Using $μ_{s} F_{N}$ as the static friction on an object that is not about to slip. Static friction is only as large as it needs to be. (Forces on Inclines and Friction)
Pointing friction down the slope for an object sliding down. Friction always opposes the relative sliding. (Forces on Inclines and Friction)
Forgetting that an angled pull or push changes the normal force, and so changes friction. (Forces on Inclines and Friction)
Setting the tension equal to the hanging mass's weight. That is only true when nothing accelerates. (Connected Objects and Frames of Reference)
Dividing the net force by just one mass instead of the total mass of everything that accelerates together. (Connected Objects and Frames of Reference)
Including tension in the system's net force. Inside a system the tension forces cancel in pairs. (Connected Objects and Frames of Reference)
Thinking an elevator moving downward must make the scale read less. The reading depends on the direction of the acceleration, not the velocity. (Connected Objects and Frames of Reference)
Adding an apparent force, such as a forward push on a braking passenger, to a free-body diagram drawn in the ground frame. (Connected Objects and Frames of Reference)
Drawing a separate 'centripetal force' arrow on a free-body diagram in addition to the real forces. The centripetal force is the net inward result of those real forces. (Uniform Circular Motion)
Saying there is no acceleration because the speed is constant. The direction of the velocity changes, so the object accelerates toward the centre. (Uniform Circular Motion)
Drawing a 'centrifugal force' pointing outward in the ground frame. No such force acts; the object's inertia makes it tend to go straight along the tangent. (Uniform Circular Motion)
Using the same equation at the top and bottom of a vertical circle. At the top gravity helps supply the centripetal force; at the bottom it works against it. (Uniform Circular Motion)
Plugging in the diameter instead of the radius, or a period in minutes instead of seconds. (Uniform Circular Motion)
Resolving forces along and perpendicular to the slope and writing $F_{N} = m g cos θ$ . On a banked curve the acceleration is horizontal, so use horizontal and vertical axes; the normal force is $m g / cos θ$ at the design speed. (Banked Curves, Rotating Frames and Artificial Gravity)
Always drawing friction up the slope. Above the design speed the car tends to slide outward, so static friction points down the slope, toward the centre. (Banked Curves, Rotating Frames and Artificial Gravity)
Forgetting to convert km/h to m/s before using $tan θ = v^{2} / (r g)$ , which makes the angle far too large or produces an impossible tangent. (Banked Curves, Rotating Frames and Artificial Gravity)
Adding a centrifugal force to a free-body diagram drawn from the ground. Centrifugal and Coriolis effects only appear when you analyse from inside the rotating frame. (Banked Curves, Rotating Frames and Artificial Gravity)
Believing sinks and toilets drain in opposite directions in each hemisphere. The Coriolis effect is far too small over centimetres and seconds; it matters for storms, ocean currents and long-range projectiles. (Banked Curves, Rotating Frames and Artificial Gravity)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Motion in Two Dimensions and Projectiles
1
At the same instant, one ball is dropped from a height and an identical ball is fired horizontally from the same height. Ignoring air resistance, which lands first?
Motion in Two Dimensions and Projectiles
2
A projectile is launched at $25 m/s$ at 60° above the horizontal. What maximum height does it reach above the launch point, in metres? Give three significant figures.
m
Motion in Two Dimensions and Projectiles
3
A boat that moves at $5.0 m/s$ relative to the water wants to arrive directly across a river flowing at $2.0 m/s$ . At what angle upstream from straight across must it head, in degrees? Give three significant figures.
°
Motion in Two Dimensions and Projectiles
4
For a projectile at its maximum height, what is the value of its vertical velocity (in m/s)?
Motion in Two Dimensions and Projectiles
5
Order the steps for solving a projectile launched at an angle from a cliff.
1. 1Combine final components if the impact velocity is needed
2. 2Use that time with the constant horizontal velocity to find the horizontal distance
3. 3Solve the vertical motion for the time of flight
4. 4Choose a positive direction and resolve the launch velocity into components
5. 5List the vertical knowns: initial vertical velocity, displacement and acceleration
Forces on Inclines and Friction
6
A $15 kg$ block slides down a frictionless incline at 35°. What is its acceleration, in $m/s^{2}$ ? Give three significant figures.
m/s²
Forces on Inclines and Friction
7
A $6.0 kg$ block slides down a 28° incline with $μ_{k} = 0.15$ . What is its acceleration, in $m/s^{2}$ ? Give three significant figures.
m/s²
Forces on Inclines and Friction
8
A $4.0 kg$ box sits at rest on a ramp. The coefficient of static friction is $0.60$ . Which expression gives the static friction acting on it?
Forces on Inclines and Friction
9
Which type of friction lets car tires grip the road without skidding when a car turns or brakes smoothly: static or kinetic?
Forces on Inclines and Friction
10
Order the steps for finding the acceleration of a block sliding down a rough incline.
1. 1Apply Newton's second law along the slope
2. 2Find the normal force from the perpendicular balance, then friction
3. 3Resolve gravity into mg sin θ and mg cos θ
4. 4Draw a free-body diagram showing gravity, normal force and friction
5. 5Tilt the axes so one lies along the slope
Connected Objects and Frames of Reference
11
An Atwood machine holds $2.0 kg$ and $2.5 kg$ masses over an ideal pulley. What is the acceleration, in $m/s^{2}$ ? Give three significant figures.
m/s²
Connected Objects and Frames of Reference
12
A $55 kg$ person stands on a scale in an elevator that accelerates downward at $2.0 m/s^{2}$ . What does the scale read, in newtons? Give three significant figures.
N
Connected Objects and Frames of Reference
13
An elevator's cable snaps and it falls freely. What does a bathroom scale inside it read?
Connected Objects and Frames of Reference
14
What name is given to a force, like the forward 'push' felt in a braking bus, that has no physical agent and appears only in an accelerating frame?
Connected Objects and Frames of Reference
15
Match each situation to the force that the free-body diagram needs.
Cart pulled across a table by a hanging mass
Crate staying put on an accelerating truck
Person on a scale in an elevator
Passenger lurching forward in a braking bus (ground frame)
Uniform Circular Motion
16
A cyclist rides around a circular track of radius $30 m$ at a constant $12 m/s$ . What is her centripetal acceleration, in $m/s^{2}$ ? Give three significant figures.
m/s²
Uniform Circular Motion
17
What is the minimum speed a roller-coaster car must have at the top of a vertical loop of radius $8.0 m$ to stay on the track, in m/s? Give three significant figures.
m/s
Uniform Circular Motion
18
A ball is whirled in a horizontal circle on a string. At the instant the string breaks, in which direction does the ball move (seen from above)?
Uniform Circular Motion
19
A passenger in a car turning left feels pushed against the right-hand door. What is really happening, as seen from the road?
Uniform Circular Motion
20
Order these objects from largest centripetal acceleration to smallest.
1. 12 m/s on a circle of radius 1 m
2. 26 m/s on a circle of radius 4 m
3. 34 m/s on a circle of radius 1 m
4. 410 m/s on a circle of radius 2 m
5. 510 m/s on a circle of radius 5 m
Banked Curves, Rotating Frames and Artificial Gravity
21
A curve of radius $150 m$ is to be banked for a design speed of $90 km/h$ with no friction needed. What angle is required, in degrees? Give three significant figures.
°
Banked Curves, Rotating Frames and Artificial Gravity
22
A space station has a radius of $250 m$ . What rotation period, in seconds, gives an apparent gravity of $9.81 m/s^{2}$ at the rim? Give three significant figures.
s
Banked Curves, Rotating Frames and Artificial Gravity
23
A car takes an icy (effectively frictionless) banked curve well below its design speed. What happens?
Banked Curves, Rotating Frames and Artificial Gravity
24
What is the name of the apparent outward force felt by someone analysing motion from inside a rotating frame?
Banked Curves, Rotating Frames and Artificial Gravity
25
Match each situation to the force that pulls or pushes the object toward the centre of its circle.
Car at the design speed on a banked curve
Car on a flat, dry curve
Astronaut on the rim of a spinning station
Rider on a rotor ride after the floor drops

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…