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Part A – Buoyant force
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| (a) 1 | \(B = W_{\text{air}} – W_{\text{water}}\) | The buoyant force \(B\) equals the difference between the weight of the object measured in air and the apparent weight when submerged. |
| (a) 2 | \(B = 17.8\,N – 16.2\,N = 1.6\,N\) | Substitute the given readings to calculate the buoyant force. |
| (a) 3 | \(\boxed{B = 1.6\,N}\) | This is the final buoyant force acting on the object in water. |
Part B – Volume of the object
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| (b) 1 | \(B = \rho_{w} g V\) | According to Archimedes’ principle, the buoyant force is equal to the weight of the displaced water where \(\rho_{w}\) is the density of water, \(g\) is gravitational acceleration, and \(V\) is the volume displaced. |
| (b) 2 | \(V = \frac{B}{\rho_{w} g}\) | Rearrange the formula to solve for the volume of the object. |
| (b) 3 | \(V = \frac{1.6}{1000 \times 9.8}\) | Substitute \(B = 1.6\,N\), \(\rho_{w} = 1000\,kg/m^3\), and \(g = 9.8\,m/s^2\). |
| (b) 4 | \(V \approx 1.63 \times 10^{-4}\,m^3\) | Compute the division \(1.6/(9800)\) to obtain the object’s volume. |
| (b) 5 | \(\boxed{V \approx 1.63 \times 10^{-4}\,m^3}\) | This is the final volume of the object. |
Part C – Density of the object
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| (c) 1 | \(W = m g\) | The weight of the object in air is the product of its mass \(m\) and gravitational acceleration \(g\). |
| (c) 2 | \(m = \frac{W}{g} = \frac{17.8}{9.8}\) | Solve for the mass by rearranging the weight formula using \(W = 17.8\,N\) and \(g = 9.8\,m/s^2\). |
| (c) 3 | \(m \approx 1.82\,kg\) | Performing the division gives the mass of the object. |
| (c) 4 | \(\rho = \frac{m}{V}\) | Density is defined as mass divided by volume. |
| (c) 5 | \(\rho = \frac{1.82}{1.63 \times 10^{-4}}\) | Substitute \(m \approx 1.82\,kg\) and \(V \approx 1.63 \times 10^{-4}\,m^3\) into the density formula. |
| (c) 6 | \(\rho \approx 1.12 \times 10^4\,kg/m^3\) | The division yields the density of the object. |
| (c) 7 | \(\boxed{\rho \approx 1.12 \times 10^4\,kg/m^3}\) | This is the final density of the object. |
Part D – Absolute pressure when object is removed
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| (d) 1 | \(p = p_{\text{atm}} + \rho_{w} g h\) | The absolute pressure at the bottom of a water column is given by the sum of atmospheric pressure \(p_{\text{atm}}\) and the hydrostatic pressure \(\rho_{w} g h\), where \(h\) is the water depth. |
| (d) 2 | Removing the object | Upon removal of the object, the water can now fill the space the ball occupied. This reduces the overall water depth \(h\). |
| (d) 4 | \(\boxed{p \text{ decreases}}\) | Thus, the hydrostatic pressure (\(\rho gh\)) decreases, due to the decrease in height of the water. Since hydrostatic pressure drops so will the absolute pressure, as given by the equation in (d) 1 |
Just ask: "Help me solve this problem."
The experimental diving rig is lowered from rest at the ocean’s surface and reaches a maximum depth of \(80\) \(\text{m}\). Initially it accelerates downward at a rate of \(0.10\) \(\text{m/s}^2\) until it reaches a speed of \(2.0\) \(\text{m/s}\), which then remains constant. During the descent, the pressure inside the bell remains constant at \(1\) atmosphere. The top of the bell has a cross-sectional area \(A = 9.0\) \(\text{m}^2\). The density of seawater is \(1025\) \(\text{kg/m}^3\).
A pump is used to send water through a hose, the diameter of which is \( 10 \) times that of the nozzle through which the water exits. If the nozzle is \( 1 \) \(\text{m}\) higher than the pump, and the water flows through the hose at \( 0.4 \) \(\text{m/s}\), what is the difference in pressure between the pump and the atmosphere?
A \(2\)-N force is used to push a small piston \(10\) \(\text{cm}\) downward in a simple hydraulic machine. If the opposite large piston rises by \(0.5\) \(\text{cm}\), what is the maximum weight the large piston can lift?
A helium-filled balloon is attached by a string of negligible mass to a small \(0.015 \ \text{kg}\) object that is just heavy enough to keep the balloon from rising. The total mass of the balloon, including the helium, is \(0.0050 \ \text{kg}\). The density of air is \(\rho_{\text{air}} = 1.29 \ \text{kg/m}^3\), and the density of helium is \(\rho_{\text{He}} = 0.179 \ \text{kg/m}^3\). The buoyant force on the \(0.015 \ \text{kg}\) object is small enough to be negligible.
The figure shows a horizontal pipe with sections with different cross-sectional areas. Small tubes extend from the top of each section. The cross-sectional area of the pipe at location C is half that at A, and the areas at A and D are the same. Water flows in the pipe from left to right. Which of the following correctly ranks the height \( h \) of the water in the tubes above the labeled locations?
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| Kinematics | Forces |
|---|---|
| \(\Delta x = v_i t + \frac{1}{2} at^2\) | \(F = ma\) |
| \(v = v_i + at\) | \(F_g = \frac{G m_1 m_2}{r^2}\) |
| \(v^2 = v_i^2 + 2a \Delta x\) | \(f = \mu N\) |
| \(\Delta x = \frac{v_i + v}{2} t\) | \(F_s =-kx\) |
| \(v^2 = v_f^2 \,-\, 2a \Delta x\) |
| Circular Motion | Energy |
|---|---|
| \(F_c = \frac{mv^2}{r}\) | \(KE = \frac{1}{2} mv^2\) |
| \(a_c = \frac{v^2}{r}\) | \(PE = mgh\) |
| \(T = 2\pi \sqrt{\frac{r}{g}}\) | \(KE_i + PE_i = KE_f + PE_f\) |
| \(W = Fd \cos\theta\) |
| Momentum | Torque and Rotations |
|---|---|
| \(p = mv\) | \(\tau = r \cdot F \cdot \sin(\theta)\) |
| \(J = \Delta p\) | \(I = \sum mr^2\) |
| \(p_i = p_f\) | \(L = I \cdot \omega\) |
| Simple Harmonic Motion | Fluids |
|---|---|
| \(F = -kx\) | \(P = \frac{F}{A}\) |
| \(T = 2\pi \sqrt{\frac{l}{g}}\) | \(P_{\text{total}} = P_{\text{atm}} + \rho gh\) |
| \(T = 2\pi \sqrt{\frac{m}{k}}\) | \(Q = Av\) |
| \(x(t) = A \cos(\omega t + \phi)\) | \(F_b = \rho V g\) |
| \(a = -\omega^2 x\) | \(A_1v_1 = A_2v_2\) |
| Constant | Description |
|---|---|
| [katex]g[/katex] | Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface |
| [katex]G[/katex] | Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex] |
| [katex]\mu_k[/katex] and [katex]\mu_s[/katex] | Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion. |
| [katex]k[/katex] | Spring constant, in [katex]\text{N/m}[/katex] |
| [katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex] | Mass of the Earth |
| [katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex] | Mass of the Moon |
| [katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex] | Mass of the Sun |
| Variable | SI Unit |
|---|---|
| [katex]s[/katex] (Displacement) | [katex]\text{meters (m)}[/katex] |
| [katex]v[/katex] (Velocity) | [katex]\text{meters per second (m/s)}[/katex] |
| [katex]a[/katex] (Acceleration) | [katex]\text{meters per second squared (m/s}^2\text{)}[/katex] |
| [katex]t[/katex] (Time) | [katex]\text{seconds (s)}[/katex] |
| [katex]m[/katex] (Mass) | [katex]\text{kilograms (kg)}[/katex] |
| Variable | Derived SI Unit |
|---|---|
| [katex]F[/katex] (Force) | [katex]\text{newtons (N)}[/katex] |
| [katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy) | [katex]\text{joules (J)}[/katex] |
| [katex]P[/katex] (Power) | [katex]\text{watts (W)}[/katex] |
| [katex]p[/katex] (Momentum) | [katex]\text{kilogram meters per second (kgm/s)}[/katex] |
| [katex]\omega[/katex] (Angular Velocity) | [katex]\text{radians per second (rad/s)}[/katex] |
| [katex]\tau[/katex] (Torque) | [katex]\text{newton meters (Nm)}[/katex] |
| [katex]I[/katex] (Moment of Inertia) | [katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex] |
| [katex]f[/katex] (Frequency) | [katex]\text{hertz (Hz)}[/katex] |
General Metric Conversion Chart
Example of using unit analysis: Convert 5 kilometers to millimeters.
Start with the given measurement: [katex]\text{5 km}[/katex]
Use the conversion factors for kilometers to meters and meters to millimeters: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]
Perform the multiplication: [katex]\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}[/katex]
Simplify to get the final answer: [katex]\boxed{5 \times 10^6 \, \text{mm}}[/katex]
Prefix | Symbol | Power of Ten | Equivalent |
|---|---|---|---|
Pico- | p | [katex]10^{-12}[/katex] | |
Nano- | n | [katex]10^{-9}[/katex] | |
Micro- | µ | [katex]10^{-6}[/katex] | |
Milli- | m | [katex]10^{-3}[/katex] | |
Centi- | c | [katex]10^{-2}[/katex] | |
Deci- | d | [katex]10^{-1}[/katex] | |
(Base unit) | – | [katex]10^{0}[/katex] | |
Deca- or Deka- | da | [katex]10^{1}[/katex] | |
Hecto- | h | [katex]10^{2}[/katex] | |
Kilo- | k | [katex]10^{3}[/katex] | |
Mega- | M | [katex]10^{6}[/katex] | |
Giga- | G | [katex]10^{9}[/katex] | |
Tera- | T | [katex]10^{12}[/katex] |
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