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Physics - Hard

1. A particle moves along the x-axis such that its position is given by \(x(t) = 2t^3 - 9t^2 + 12t\). At what time is its acceleration zero?

\(t = 1.5\text{ s}\)
\(t = 1\text{ s}\)
\(t = 2\text{ s}\)
\(t = 0.5\text{ s}\)

2. A block of mass \(m\) slides down an inclined plane of angle \(\theta\) with constant velocity. What is the coefficient of kinetic friction \(\mu_k\)?

\(\tan\theta\)
\(\sin\theta\)
\(\cos\theta\)
\(\cot\theta\)

3. Two masses \(m_1\) and \(m_2\) (\(m_1 > m_2\)) connected by a light string pass over a frictionless pulley. What is the magnitude of acceleration of the system?

\(\frac{m_1 - m_2}{m_1 + m_2}g\)
\(\frac{m_1 + m_2}{m_1 - m_2}g\)
\(\frac{2m_1 m_2}{m_1 + m_2}g\)
\(\frac{m_1 m_2}{m_1 + m_2}g\)

4. The moment of inertia of a uniform solid sphere of mass \(M\) and radius \(R\) about its diameter is:

\(\frac{2}{5}MR^2\)
\(\frac{2}{3}MR^2\)
\(\frac{1}{2}MR^2\)
\(\frac{7}{5}MR^2\)

5. An ideal monoatomic gas undergoes an adiabatic expansion such that its volume doubles. By what factor does its pressure change?

\(2^{-5/3}\)
\(2^{-3/5}\)
\(2^{5/3}\)
\(2^{-7/5}\)