1

Useful Formulae and Constants

Trigonometry

sinθ =yr cosθ =

xr

tanθ =yx

sin cos2 2 1θ θ+ =

sin sin cos2 2θ θ θ=

cos cos sin2 2 2θ θ θ= −

x y r2 2 2+ =

Quadratic Formula

ax bx c2 0+ + =

xb b ac

a=− ± −2 4

2

Circle

A r= π 2

C r= 2π

Sphere

A r= 4 2π

V r=43

3π

Units 1 in 2.54 cm=

1 mile km= 161.

1 liter cm3= 1000

One-Dimensional Motion

txvvvΔΔ

=+

=2

01

avt

=ΔΔ

atvtv += 0)(

200 2

1)( attvxtx ++=

( )v v a x x202

02= + −

Forces

F a∑ = m

nkkf FF μ= F Fsf s n≤ μ

kxFs −=

Uniform Circular Motion

avrc =2

Trv π2

= T f=

1

Gravity

F Gm m

r= 1 2

2

32

2 4 RGM

T ⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

G = × −6 67 10 11.Nmkg

2

2

g = 9 80.ms2

2

Useful Formulae and Constants

Scalar Product

zzyyxx BABABABA ++=•rr

Cross Product

iBABABA yzzyˆ)( −=×

rr

jBABA zxxzˆ)( −+

kBABA xyyxˆ)( −+

Work, Energy, and Power

θcosFddFW =•=rr

2

21 mvKE =

PE mgygrav = (near surface)

RMmGPEgrav −= (planetary)

2

21 kxPEelastic =

W KE PENC = +Δ Δ

vFt

WP rr•=

Δ==

TimeWork

Momentum and Impulse p v= m

∫=Δ=Δ= dtFtJr

Fp

Conservation of Momentum '' 22112211 vvvv mmmm +=+

1-Dimensional Elastic Collision

( )'' 2121 vvvv −−=−

Rotational Kinematics

1 2 3600rev rad= =π

,tΔ

Δ=

θω tΔΔ

=ωα

,rs θ= v r= ω , a rT= α

fT

= =1

2ωπ

ω ω α= +0 t

θ θ ω α= + +0 021

2t t

( )ω ω α θ θ202

02= + −

Rotational Dynamics

Torque: Frrr

×=τ magnitude: θτ sinrF= direction: right-hand rule

∑= ,2iirmI ∫= dmrI 2

2

21 ωIKE = (fixed axis)

22

21

21 ωcmcm ImvKE += (moving axis)

,prL rr×= ωIL =

∑ ==dtdLIατ

τω=P

Mark Reeves - Physics 21 – Spring 2008 1exam1mer

A car travels a certain distance along a straight road (in the+x direction). The distance the car travels as a function oftime is shown in the figure below. When the car changesspeed it does so uniformly.

4 pt Select the appropriate choice for each statement.

0 20 40 60 80

Time (s)

0

200

400

600

800

1000

Distance (m)

. The x-acceleration is ..... zero at 15 sec.1. A© True B© False C© Greater than

D© Less than E© Equal to

. The x-acceleration is ..... zero at 48 sec .2. A© True B© False C© Greater than

D© Less than E© Equal to

. The speed at 46 sec is ..... the speed at 73 sec.3. A© True B© False C© Greater than

D© Less than E© Equal to

. At 73 sec, the speed is as high as it gets.4. A© True B© False C© Greater than

D© Less than E© Equal to

. The speed is ..... zero at 20 sec.5. A© True B© False C© Greater than

D© Less than E© Equal to

. The acceleration magnitude at 15 sec is ..... the accelerationmagnitude at 37 sec.

6. A© True B© False C© Greater than

D© Less than E© Equal to

4 pt Using the graph above, calculate the average speed of

the car for the period from 0 to 70.0 s(in m/s)

7.A© 9.79 B© 1.22× 101C© 1.53× 101

D© 1.91× 101E© 2.39× 101

F© 2.99× 101

G© 3.74× 101H© 4.67× 101

1

4 pt A Boeing 727-200 jet taking off from Lansing airport

accelerates from rest for 31.0 s before leaving the ground. Itsacceleration is 2.21 m/s2. Assuming that the acceleration isconstant, calculate the plane’s speed at take off.

(in m/s)

8.A© 2.247× 101B© 3.259× 101

C© 4.725× 101D© 6.851× 101

E© 9.934× 101F© 1.440× 102

G© 2.089× 102H© 3.028× 102

4 pt What is the take off speed in mi/hr?

9.A© 1.535× 102B© 1.796× 102

C© 2.101× 102D© 2.458× 102

E© 2.876× 102F© 3.365× 102

G© 3.937× 102H© 4.606× 102

4 pt To save fuel, some truck drivers try to maintain a con-

stant speed when possible. A truck traveling at 25.0 km/hrapproaches a car stopped at the red light. When the truck is154.8 meters from the car the light turns green and the carimmediately begins to accelerate at 2.1 m/s2. How close doesthe truck come to the car assuming the truck does not slowdown?

(in m)

10.A© 4.580× 101B© 6.092× 101

C© 8.102× 101D© 1.078× 102

E© 1.433× 102F© 1.906× 102

G© 2.535× 102H© 3.372× 102

4 pt How far from the stop light has the car travelled when

the truck reaches its closest distance?(in m)

11.A© 5.237 B© 6.127

C© 7.169 D© 8.388

E© 9.814 F© 1.148× 101

G© 1.343× 101H© 1.572× 101

2

Mark Reeves - Physics 21 – Spring 2008 3exam1mer

4 pt The velocity vector V1 has a magnitude of 3.0 m/s and

is directed along the +x-axis. The velocity vector V2 has amagnitude of 2.0 m/s. The sum of the two is V3, so that V3

= V1+V2

. The magnitude of V3 can be 5.0 m/s12. A© True B© False

. The magnitude of V3 can be 6.0 m/s13. A© True B© False

. The x-component of V3 can be -1.0 m/s14. A© True B© False

. The magnitude of V3 can be 2.0 m/s15. A© True B© False

. The magnitude of V3 can be -2.0 m/s16. A© True B© False

. The magnitude of V3 can be 0.017. A© True B© False

4 pt

A boy standing in aditch throws a base-ball upward towardhis father. The ballleaves his hand atground level, with aninitial speed of 13.0m/s,at an angle of θ =59◦, from the horizon-tal.

θ

The boy’s father reaches up and catches the ball over his head,at a height of 2.0 m above the ground. The father catchesthe ball on its way down (as shown in the Figure). Calculatehow long the ball is in the air. ( g = 9.81 m/s2)(in s)

18.A© 2.075 B© 2.428 C© 2.841 D© 3.324

E© 3.889 F© 4.550 G© 5.324 H© 6.229

3

4 pt A 2.21 kg mass is suspended from a string which is

pulled upward. The mass accelerates upwards with an accel-eration of 2.50 m/s2. What is the tension in the string?(in N)

19.A© 2.019 B© 2.927

C© 4.244 D© 6.154

E© 8.924 F© 1.294× 101

G© 1.876× 101H© 2.721× 101

4 pt

. The net force which acts on an object which maintains aconstant velocity is zero.20. A© True B© False

. If two objects have the same acceleration, they are underthe influence of equal forces.21. A© True B© False

. If a net force acts on an object, the object’s velocity willchange.22. A© True B© False

. If two objects are under the influence of equal forces, theyhave the same acceleration.23. A© True B© False

. If an object’s speed does not change, no net force is actingon the object.24. A© True B© False

4 pt M1 and M2 are two masses connected as shown.

θ

M 1

2 M

The pulley is light and frictionless. Find the mass M1, giventhat M2 (7.00 kg) is moving downwards and accelerates down-wards at 2.43 m/s2, that θ is 15.0◦, and that µk is 0.410.(in kg)

25.A© 3.16 B© 3.58 C© 4.04 D© 4.57

E© 5.16 F© 5.83 G© 6.59 H© 7.45

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MSU Licensed under GNU General Public License

4

26. When a spherical object moves through water, there is a drag force that depends on the product of the radius of the object and its velocity, i.e., Fdrag = 6πηRv, where η is the viscosity, a constant equal to 1 x 10-3 for water (in SI units). Determine the dimension of η. A Ns/m2 B Ns2/m2 C Ns/m3

D Ns/m 27. Assume that an amoeba is a spherical object with a radius of 100 microns and that it is composed mostly of water (ρ = 1 g/mL). What is its mass? 28. An amoeba with cross-sectional radius of 100 microns moves at a constant velocity of 100 microns per second. If the amoeba dies, it will stop. Which statement accurately describes the stopping of the amoeba

A. It will stop instantly B. Its velocity will decrease linearly with time. C. Its velocity will decrease at a faster rate as the amoeba slows D. Its velocity will decrease at a slower rate as the amoeba slows

28. Make a reasoned, numerical estimate the distance that the amoeba will travel. Tell what assumptions you are making in your estimate.

29. Is this an overestimate or an underestimate or is it dead-on – tell why.