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Finding tension in the string

Step Formula Derivation Reasoning
1 \tan(\theta) = \frac{R}{h} The angle \theta is between the string and the vertical axis. h is the vertical distance from the ball to the pivot.
2 h = L \cos(\theta) From the geometry of the cone.
3 \tan(\theta) = \frac{R}{L \cos(\theta)} Substituting the expression for h.
4 \sin(\theta) = \frac{R}{\sqrt{R^2 + h^2}} From the right triangle formed by the string, h, and R.
5 F_{T} \sin(\theta) = F_{c} The horizontal component of tension provides the centripetal force (F_{c}).
6 F_{c} = m \frac{v^2}{R} Centripetal force formula. v is the velocity of the ball.
7 F_{T} \cos(\theta) = mg The vertical component of tension balances gravity.
8 F_{T} = \frac{mg}{\cos(\theta)} Isolating F_{T} in the vertical balance.
9 F_{T} = \frac{mg}{\cos(\theta)} = \frac{mg}{\sqrt{1 – \sin^2(\theta)}} Using \cos(\theta) = \sqrt{1 – \sin^2(\theta)}.
10 F_{T} = \frac{mg}{\sqrt{1 – \left(\frac{R}{\sqrt{R^2 + h^2}}\right)^2}} Substituting \sin(\theta).
11 \boxed{F_{T} = \frac{mg}{\sqrt{1 – \frac{R^2}{R^2 + L^2 \cos^2(\theta)}}}} Final expression for tension, substituting h = L \cos(\theta).

Finding period of the pendulum

Step Formula Derivation Reasoning
1 F_{c} = m \frac{v^2}{R} Centripetal force formula.
2 F_{T} \sin(\theta) = m \frac{v^2}{R} The horizontal component of tension provides the centripetal force.
3 v = R \omega Relationship between linear velocity and angular velocity (\omega).
4 m \frac{(R \omega)^2}{R} = F_{T} \sin(\theta) Substituting v with R \omega.
5 \omega^2 = \frac{F_{T} \sin(\theta)}{mR} Isolating \omega^2.
6 \omega = \sqrt{\frac{g}{R \tan(\theta)}} Using F_{T} \sin(\theta) = mg \sin(\theta) and simplifying.
7 T = \frac{2\pi}{\omega} Period (T

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1. F_{T} = \frac{mg}{\sqrt{1 – \frac{R^2}{R^2 + L^2 \cos^2(\theta)}}}
2. T = \frac{2\pi}{\omega}

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KinematicsForces
\Delta x = v_i t + \frac{1}{2} at^2F = ma
v = v_i + atF_g = \frac{G m_1m_2}{r^2}
a = \frac{\Delta v}{\Delta t}f = \mu N
R = \frac{v_i^2 \sin(2\theta)}{g}
Circular MotionEnergy
F_c = \frac{mv^2}{r}KE = \frac{1}{2} mv^2
a_c = \frac{v^2}{r}PE = mgh
KE_i + PE_i = KE_f + PE_f
MomentumTorque and Rotations
p = m v\tau = r \cdot F \cdot \sin(\theta)
J = \Delta pI = \sum mr^2
p_i = p_fL = I \cdot \omega
Simple Harmonic Motion
F = -k x
T = 2\pi \sqrt{\frac{l}{g}}
T = 2\pi \sqrt{\frac{m}{k}}
ConstantDescription
gAcceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface
GUniversal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2
\mu_k and \mu_sCoefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion.
kSpring constant, in \text{N/m}
M_E = 5.972 \times 10^{24} , \text{kg} Mass of the Earth
M_M = 7.348 \times 10^{22} , \text{kg} Mass of the Moon
M_M = 1.989 \times 10^{30} , \text{kg} Mass of the Sun
VariableSI Unit
s (Displacement)\text{meters (m)}
v (Velocity)\text{meters per second (m/s)}
a (Acceleration)\text{meters per second squared (m/s}^2\text{)}
t (Time)\text{seconds (s)}
m (Mass)\text{kilograms (kg)}
VariableDerived SI Unit
F (Force)\text{newtons (N)}
E, PE, KE (Energy, Potential Energy, Kinetic Energy)\text{joules (J)}
P (Power)\text{watts (W)}
p (Momentum)\text{kilogram meters per second (kgm/s)}
\tau (Torque)\text{newton meters (Nm)}
I (Moment of Inertia)\text{kilogram meter squared (kgm}^2\text{)}
f (Frequency)\text{hertz (Hz)}

General Metric Conversion Chart

Example of using unit analysis: Convert 5 kilometers to millimeters.

1. Start with the given measurement: \text{5 km}

2. Use the conversion factors for kilometers to meters and meters to millimeters: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}

3. Perform the multiplication: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}

4. Simplify to get the final answer: \boxed{5 \times 10^6 \, \text{mm}}

Prefix

Symbol

Power of Ten

Equivalent

Pico-

p

10^{-12}

Nano-

n

10^{-9}

Micro-

µ

10^{-6}

Milli-

m

10^{-3}

Centi-

c

10^{-2}

Deci-

d

10^{-1}

(Base unit)

10^{0}

Deca- or Deka-

da

10^{1}

Hecto-

h

10^{2}

Kilo-

k

10^{3}

Mega-

M

10^{6}

Giga-

G

10^{9}

Tera-

T

10^{12}

1. Some answers may be slightly off by 1% depending on rounding, etc.
2. Answers will use different values of gravity. Some answers use 9.81 m/s2, and other 10 m/s2 for calculations.
3. Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
4. Bookmark questions that you can’t solve so you can come back to them later.
5. Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

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