Question

1. (4 points) a constant force is applied to a crate resting on the floor. the crate accelerates at 10.0 m/s². what is the acceleration if: a. the force is doubled. b. the mass is quadrupled c. the mass is doubled, and the force is quadrupled d. both the force and the mass are doubled

Explanation:

Step1: Recall Newton's second law

Newton's second law is \(F = ma\), where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. We can rewrite it as \(a=\frac{F}{m}\). Initially, \(a_0 = 10.0\ m/s^{2}=\frac{F_0}{m_0}\)

Step2: Solve part (a)

If the force is doubled (\(F_1 = 2F_0\)) and the mass remains \(m_1=m_0\), then \(a_1=\frac{F_1}{m_1}\). Substitute \(F_1 = 2F_0\) and \(m_1=m_0\) into the formula: \(a_1=\frac{2F_0}{m_0}\). Since \(\frac{F_0}{m_0}=a_0 = 10.0\ m/s^{2}\), then \(a_1 = 2\times10.0\ m/s^{2}=20.0\ m/s^{2}\)

Step3: Solve part (b)

If the mass is quadrupled (\(m_2 = 4m_0\)) and the force remains \(F_2=F_0\), then \(a_2=\frac{F_2}{m_2}\). Substitute \(F_2 = F_0\) and \(m_2 = 4m_0\) into the formula: \(a_2=\frac{F_0}{4m_0}\). Since \(\frac{F_0}{m_0}=a_0 = 10.0\ m/s^{2}\), then \(a_2=\frac{10.0\ m/s^{2}}{4}=2.5\ m/s^{2}\)

Step4: Solve part (c)

If the mass is doubled (\(m_3 = 2m_0\)) and the force is quadrupled (\(F_3 = 4F_0\)), then \(a_3=\frac{F_3}{m_3}\). Substitute \(F_3 = 4F_0\) and \(m_3 = 2m_0\) into the formula: \(a_3=\frac{4F_0}{2m_0}\). Since \(\frac{F_0}{m_0}=a_0 = 10.0\ m/s^{2}\), then \(a_3 = 2\times10.0\ m/s^{2}=20.0\ m/s^{2}\)

Step5: Solve part (d)

If both the force and the mass are doubled (\(F_4 = 2F_0\) and \(m_4 = 2m_0\)), then \(a_4=\frac{F_4}{m_4}\). Substitute \(F_4 = 2F_0\) and \(m_4 = 2m_0\) into the formula: \(a_4=\frac{2F_0}{2m_0}\). Since \(\frac{F_0}{m_0}=a_0 = 10.0\ m/s^{2}\), then \(a_4=10.0\ m/s^{2}\)

Answer:

a. \(20.0\ m/s^{2}\)
b. \(2.5\ m/s^{2}\)
c. \(20.0\ m/s^{2}\)
d. \(10.0\ m/s^{2}\)