Question

policies
current attempt in progress
an object is oscillating in simple harmonic motion with an amplitude a and an angular frequency ω. what should you do to increase the maximum speed of the motion?
○ reduce a by 10% and increase ω by 10%.
○ reduce both a and ω by 10%.
○ increase both a and ω by 10%.
○ increase a by 10% and reduce ω by 10%.

Explanation:

Step1: Recall the formula for maximum speed in SHM

The maximum speed \(v_{max}\) in simple - harmonic motion is given by \(v_{max}=A\omega\).

Step2: Analyze each option

• Option 1: Reduce \(A\) by \(10\%\) and increase \(\omega\) by \(10\%\)

Let \(A_1 = 0.9A\) and \(\omega_1=1.1\omega\). Then \(v_{max1}=A_1\omega_1=(0.9A)(1.1\omega)=0.99A\omega<v_{max}\)

• Option 2: Reduce both \(A\) and \(\omega\) by \(10\%\)

Let \(A_2 = 0.9A\) and \(\omega_2 = 0.9\omega\). Then \(v_{max2}=A_2\omega_2=(0.9A)(0.9\omega)=0.81A\omega<v_{max}\)

• Option 3: Increase both \(A\) and \(\omega\) by \(10\%\)

Let \(A_3 = 1.1A\) and \(\omega_3=1.1\omega\). Then \(v_{max3}=A_3\omega_3=(1.1A)(1.1\omega)=1.21A\omega > v_{max}\)

• Option 4: Increase \(A\) by \(10\%\) and reduce \(\omega\) by \(10\%\)

Let \(A_4 = 1.1A\) and \(\omega_4 = 0.9\omega\). Then \(v_{max4}=A_4\omega_4=(1.1A)(0.9\omega)=0.99A\omega<v_{max}\)

Answer:

Increase both \(A\) and \(\omega\) by \(10\%\) (the option corresponding to this description in the original multiple - choice list).