1. A cart of weight 25 N is released at the top of an inclined plane of length 1m, which makes an angle of 30° with the ground. It rolls down the plane and hits another cart of weight 40 N at the bottom of the incline. Calculate the speed of the first cart at the bottom of the incline and the speed at which both carts move together after the impact.
2. a) Why are cars made with bumpers that can be pushed in during a crash?
b) A crane operated by an electric motor has a mass of 500 kg. It raises a load of 300 kg vertically at a steady speed of 0.2 ms-1. Assume resistance to be constant at 1200 N.
What is the power required to raise the load?
3. a) A grinding wheel starts from rest and attains a constant angular acceleration of 5.0 rad s-2. Calculate the acceleration at a point 1.0 m from the axis at t = 6.0 s.
b) An ice skater is spinning about the vertical axis passing through her body. How can her angular momentum be changed? How can her angular velocity be changed without changing her angular momentum?
4. A block of mass 3 kg starts from rest and slides down a surface, which corresponds to a quarter of a circle of 1.6 m radius.
a) If the curved surface is smooth, calculate the speed of the block at the bottom.
b) If the block's speed at the bottom is 4 ms-1, what is the energy dissipated by friction as it slides down?
c) After the block reaches the horizontal surface with a speed of 4 ms-1, it stops after travelling a distance of 3 m from the bottom. Find the frictional force acting on the horizontal surface due to the block.
5. The potential energy (in J) of a system in one dimension is given by:
U(x) = 5 - x + 3x2 - 2x3
What is the work done in moving a particle in this potential from x = 1 m to x = 2 m? What is the force on a particle in this potential at x = 1 m and x = 2 m? Locate the points of stable and unstable equilibrium for this system.
6. Three point masses of 3 kg each have the following position vectors:
→r1(t) = (2t+3t2)mi^ + tmk^; r2(t) = 4t2mj^ + 3mk^;r3(t) = (3t-1)m^i + 3t2 mj^
Determine the velocity and acceleration of the centre of mass of the system.
7. What are the conditions for the mechanical equilibrium of a rigid body?
A ladder is placed against a wall making an angle θ with the floor. The wall is frictionless but the coefficients of static and kinetic friction for the floor are µs and µk, respectively.
Obtain an expression for the smallest value of θ for the ladder to not slip.
8. Particle 1 of mass 3m initially moving with a speed v0 in the positive x-direction collides with particle 2 of mass m moving in the opposite direction with an unknown speed v. After collision, particle 1 moves along the negative y-direction with speed v0/2 and particle 2 moves with a speed v′ in a direction making an angle of 45° with the positive x-direction.
Determine v and v′ in units of v0. Is the collision elastic?
9. The planet Jupiter has an elliptical orbit with e = 0.05 and a semi-major axis of 7.8 x1011 Calculate the energy of the planet, perihelion and aphelion distances and the speed of the planet at these points.
10. a) At what angular speed must a centrifuge rotate if a particle placed 7.0 cm from its axis of rotation is to experience an acceleration of 1000g? (Take g = 10 ms-2)
b) Explain why hurricanes spin counter clockwise in the Northern hemisphere and clockwise in the Southern hemisphere.