Suppose that a string of length \(L\) and mass \(m\) is under tension \(F\). (a) Show that \(\sqrt{F L} m\) has units of speed. (b) Show that there is no other combination of \(L, m,\) and \(F\) with units of speed. [Hint: Of the dimensions of the three quantities $L, m, \text { and } F, \text { only } F \text { includes time. }]$ Thus, the speed of transverse waves on the string can only be some dimensionless constant times \(\sqrt{F L / m}.\)

Short Answer

Expert verified
Question: Verify if the expression \(\sqrt{F L} m\) has units of speed and show there is no other combination of \(L\), \(m\), and \(F\) with units of speed. Answer: The given expression \(\sqrt{F L} m\) has units of speed, which are meters per second (m/s). There is no other combination of \(L\), \(m\), and \(F\) with units of speed, as we can't perfectly cancel out the dimensions of mass and length to achieve the correct speed units.

Step by step solution

01

Identify the base units of each given variable

We are given the following quantities: \(L\) is length, \(m\) is mass, and \(F\) is tension. These quantities have the base units of: - \(L\) (length) - meters (m) - \(m\) (mass) - kilograms (kg) - \(F\) (tension) - Newtons (N) It's important to note that 1 Newton is equivalent to 1 kg*m/s², as it is a derived unit for force.
02

Break down the given expression's units

The given expression is \(\sqrt{F L} m\). To determine the units of this expression, we need to analyze the units of each part, and combine them accordingly. In the square root, we have \(F\) (Newtons) multiplied by \(L\) (meters). So, the units inside the square root would be N*m, which is equivalent to kg*m²/s². We also have a mass (\(m\)) outside the square root, which has units of kg. Now, we'll derive the units of the expression by multiplying the units of all parts: \(\sqrt{kg \cdot m^2 / s^2} \cdot kg\)
03

Find the combined units

To simplify the units of the expression, we must take the square root of the units inside the square root: \(\sqrt{kg \cdot m^2 / s^2} = m/s\) Now, when we multiply this by the mass units (kg) outside the square root, we get: \(m/s \cdot kg = (m * kg) / s\) So, the given expression has units of speed, which are meters per second (m/s). #b# Show there is no other combination of \(L, m, F\) with units of speed
04

Analyze the dimensions

To show that there is no other combination of \(L, m, F\) with units of speed, we can analyze the dimensions of these quantities and find other combinations with the dimensions of speed. Recall that dimensions (the base units) of the three quantities are: - \(L\) (length) - meters (m) - \(m\) (mass) - kilograms (kg) - \(F\) (tension) - Newtons (N) And units of speed are m/s.
05

Find combinations with units of speed

Since a speed requires both distance and time, any correct combination needs to include \(F\) - only \(F\) contains a time dimension, as N = kg * m/s². To find other combinations, we must create a combination of \(L, m, F\) such that the final dimensions come out as m/s. The only way to do this is by using the dimensions of mass and length using powers: \(m^aL^bF^c\) Here, \(a, b, c\) are the powers to cancel out the incorrect dimensions and ultimately have m/s.
06

Try to cancel out incorrect dimensions

We can now solve for the powers of \(a, b\), and \(c\). Using the base units of the given quantities, we have: \(kg^a \cdot m^b \cdot \\frac{kg^c \cdot m^{3c}}{s^{2c}}\) To have units of speed, we need to balance both the mass and length dimensions and then make sure no mass dimension remains. If we try to cancel mass dimension with our given quantities, we can observe that they can't be canceled perfectly. Hence, we can't get a speed unit without this given combination. So, the given expression \(\sqrt{F L} m\) is the only combination of \(L, m, F\) with units of speed.

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