Classical Mechanics Answers

2. A mass 𝑚 rests on a frictionless horizontal table and is connected to rigid supports via two identical springs each of relaxed length 𝑙0and spring constant 𝑘, as shown in Fig. 2. Each spring is stretched to a length 𝑙 considerably greater than 𝑙0. Horizontal displacements of 𝑚 from its equilibrium position are labeled 𝑥 (along AB) and 𝑦 (perpendicular to AB). (a) Find the angular frequencies of the normal modes for longitudinal oscillations of small amplitude. (b) Find the angular frequencies of the normal modes for transverse oscillations, assuming 𝑦 << 𝑙. (c) In terms of 𝑙 and 𝑙0, calculate the ratio of the period of oscillation along 𝑥 and 𝑦. (d) If at 𝑡 = 0 the mass 𝑚 is released from the point 𝑥 = 𝑦 = 𝐴0with zero velocity, what are its 𝑥 and 𝑦 coordinates at any later time 𝑡? (e) Draw picture of the resulting path of 𝑚 under the conditions of part (d) if = 9𝑙0/5.

1) A 90.0-N box is pulled 10.0 m along a level surface by a rope. If the rope makes an angle of 20.0° with the surface, and the force exerted through the rope is 75.0 N, the work is done on the box is ___J.

2) A 25.0-N object is held 2.10 m above the ground. The potential energy of the object with respect to the ground is ___J.

3) The kinetic energy of a 20.0-N object is 5.00 × 10² J. The speed of this object is ___m/s.

4) A 10.0-N object is accelerated uniformly from rest at a rate of 2.5 m/s². The kinetic energy of this object after it has accelerated a distance of 15.0 m is ___J.

A compact car can climb a hill in 10 s. The top of the hill is 30 m higher than the bottom, and the car’s mass is 1,000 kg What is the power output of the car?

a sphere of radius R=40 and mass M=7.2kg. calculate the moment of inertia about the axis of rotation at a distance of R from the centre of sphere.

A 10 kg block is pushed 20.0 m up the sloping surface of a plane inclined at an angle of 37^o to the horizontal by a constant force F of 110.0 N acting parallel to the plane as shown in figure 2 below. The coefficient of kinetic friction between the block and plane is 0.25. Calculate work done by friction

A mass-spring system undergoes simple harmonic motion on a frictionless surface with amplitude 1.00 (meters) and angular frequency ω=

ω= 6.8 (rad/s).

Calculate the speed of the mass at the point where 1/6 of the total energy is kinetic energy.

Two crossed belts on pulleys of diameters 3.6 m and 2.4 m connect two parallel shafts with centres 4.2 meters apart. The maximum tension in the belts is limited to 1200 N and friction between the belts and the pulley, μ = 0.26. The smaller pulley has a speed of 300 rev/min.

2.1. Find the power that can be transmitted. (8)

2.2. What would be transmitted if open belts were used. (8)

A winding drum with a mass 400 kg and a 2 m effective diameter raises a cage of mass 800 kg by a light rope. The radius of gyration of the cage is of 1 m. A balance mass of 500 kg is also used as a counterweight. The resistance to the motion of both cage and balance mass is 0.5 N/kg. The drum is driven by an electric motor through a reduction gear system with a ratio of 4:1. If the cage must accelerate upwards at 2 m/s2, calculate the torque on the motor if the efficiency is 75%.