Section 2: Mechanics II
1. Gravitational Dependence
A simple pendulum has a period of 4 seconds on Earth. What would its period be on the Moon, where the gravitational acceleration is about 1/6th of Earth's?
What is the required length of a simple pendulum to have a period of exactly 1 second on Earth?
2. Harmonic Motion
A 10 kg mass is attached to a spring and oscillates according to the equation \(x(t) = 0.2 \cos(10\pi t)\) (in meters). What is the spring constant \(k\)? What is the total mechanical energy of the system?
3. Conservation of Energy
A pendulum with a length of 1.0 meter is released from an initial angle of \(15^\circ\). What is the speed of the pendulum bob at the bottom of its swing?
4. Energy & Momentum
A 0.5 kg block slides down a frictionless track from a height of 3.0 m. At the bottom, it collides and sticks to a 1.5 kg block, which is initially at rest. What is the speed of the combined mass just after the collision?
5. Inelastic Collision
A 70 kg runner moving at \(3 \text{ m/s}\) jumps onto a 140 kg stationary cart. What is the final speed of the cart with the runner? Is kinetic energy conserved in this collision? Explain.
6. Energy Dissipation
A tennis ball is dropped from a height of \(2.0\) m. After each bounce, it loses 30% of its mechanical energy. To what height does it rise after the second bounce?
7. Dynamics with Friction
A 5 kg block is placed on a 10 kg block. A horizontal force of 45 N is applied to the 10 kg block, and the 5 kg block is tied to the wall. The coefficient of kinetic friction between all moving surfaces is 0.2. Find the acceleration of the 10 kg block.
8. Work of a variable force
Given a one-dimensional force:
- Write down the equation of motion and solve it.
- Calculate the work done during the displacement from \(0\) to \(x_0\).
- Interpret the result as potential energy.
- Verify the relationship \(F = -\frac{dU}{dx}\).
- Draw the graph of \(F(x)\) and \(U(x)\).
9. Vertical throw with drag
We have the equation of motion:
with initial conditions \(v(0)=v_0\), \(x(0)=10\).
- Solve the equation by analytical methods.
- Determine the maximum height.
- Compare with the case without drag.
- Perform a numerical simulation using HTML or Pythyon.
10. Force field and power
In a certain force field, the equations of motion of a particle with mass \(m=0.5\) kg are as follows:
Find the time dependence of: the particle's velocity, the particle's momentum, the particle's acceleration, the force acting on the particle, and the power transferred by the field to the particle.
11. Dynamics with a time-dependent force
A particle of mass \(m=3\) kg moves in a force field \(F\) dependent on time in the following way:
Assuming initial conditions \(r_0=(5,2,-3)\) m, \(v_0=(2,0,1)\) m/s, find the dependence of the particle's position and velocity on time.
12. Work and energy with a constant force
A constant force acts on a body of mass \(m = 2\ \mathrm{kg}\):
The body starts with an initial velocity \(\vec v(0) = (1, -1)\ \mathrm{\frac{m}{s}}\) from the point \(\vec r(0)=(0,0)\ \mathrm{m}\). * Determine \(\vec a(t)\). * Determine \(\vec v(t)\). * Determine \(\vec r(t)\). * Draw the trajectory of the motion. * Calculate the work done by the force at time \(t=3\ \mathrm{s}\). * Check the consistency with the work-energy theorem.