## Friday, November 11, 2016

### Laws of Motion Problems with Solutions Five

We are solving problems about Newton's laws of motion. We deal regarding force and its implications. Force do try to change the state of the body or will change the state of the body. Momentum is the physical quantity that explains the transfer of kinetic energy at the time of collision. It is the way of interaction of the body with other body. During elastic collision, both momentum and kinetic energy is conserved where as in in elastic collision only linear momentum is conserved but kinetic energy is not conserved. In the case of inelastic collision the remaining kinetic energy is converted into light,sound and heat.

Problem

When a train is starting from rest is uniformly accelerating. A plumb bob is hanging from the roof of a compartment is found to be in inclined at an angle of 45 degree with the vertical. We need to know the time taken to travel a distance of half kilometer.

Solution

The train is moving in the forward direction and the pendulum is in that train. The body that we are observing is in non inertial frame and we cannot apply laws of motion to that system. We shall apply a pseudo force on the bob to nullify the acceleration of the system.

The forces acting on the bob are the pseudo force in the horizontal direction and weight is acting in the vertical direction. The resultant position of the bob is as shown in the diagram below. We can solve the problem as shown.

Problem

A rod of length L and mass M is acted on by two unequal force as shown in the diagram. We need to know the tension in the rod at a distance from one end. The problem is as shown in the diagram below.

Solution

We can measure the effective force as the difference between the applied force and the resultant force is along the direction of the dominating force. We can also measure the acceleration of the system.

We can measure the mass of a portion of the system as the product of linear density and the length of the part of the mass that we need to measure. Linear density is the mass per unit length.We know measure the effective force as shown below.

Problem

An explosion blows a rock into three parts. Two parts go off at right angles to each other. The parts of the mass and its velocities are given below. We need to know the velocity of the third part of the body. The problem is as shown below.

Solution

The momentum of two particles are perpendicular to each other and we can measure the effective momentum using vector laws of motion. As the two parts are perpendicular, we can find the vector sum magnitude.

We can apply law of conservation of the momentum as there is no external force acting on the system. Applying an simplifying it further, we can solve the problem as shown in the diagram below.

Problem

A hammer of mass M strikes a nail of known mass with a velocity and drives a meter into a fixed block of wood. We need to measure the average resistance of the wood to the penetration of the nail.

Solution

We can apply law of conservation of the momentum. The momentum of nail alone is the momentum of system and we can measure the velocity.

We can find the kinetic energy of the system as per its definition and we can substitute the velocity that is derived in the first step. We can solve the problem as shown in the diagram below.

Problem

It is given in the problem that two blocks are connected by a light spring of known force constant. The system is at rest on a smooth horizontal surface with the spring in its natural state without any compression or extension. A velocity is given to one body in the direction away from one body. During the subsequent motion, the maximum extension in the string. The problem is as shown in the diagram below.

Solution

We can find the effective mass of the system as the reduced mass concept. We can use the concept of kinetic energy with a known velocity and it has to be equated into energy stored in the spring.The problem is solved as shown in the diagram below.

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