Question

A pendulum bob 0.5kg is raised at
a height of 15cm before it is released. At the bottom of
in path, it makes a perfectly elastic collision with a mass 1kg that is connected to a horizontal spring of spring constant 1.5 N/m
Maximum compression in spring is 280
-------
x
find x .
​

Answer 1

Answer:

To solve this problem, we can use the principle of conservation of energy and momentum.

Explanation:

Initially, the pendulum bob has gravitational potential energy due to its height above the ground. When released, this energy is converted to kinetic energy as it moves towards the bottom of its path. At the bottom, the pendulum bob collides with the 1 kg mass and transfers some of its momentum and kinetic energy to it.

We know that the collision between the pendulum bob and the 1 kg mass is perfectly elastic, which means that both momentum and kinetic energy are conserved. After the collision, the pendulum bob and the 1 kg mass move together as a single system, and the kinetic energy is stored in the spring as elastic potential energy.

We can use the formula for the elastic potential energy of a spring, which is given by:

Elastic potential energy = (1/2) k x^2

Where k is the spring constant, and x is the compression in the spring. We are given that the maximum compression in the spring is 0.28 m (since the units are not provided, I am assuming it is meters), so we can substitute this value into the formula and solve for x:

Elastic potential energy = (1/2) * 1.5 * (0.28)^2

= 0.0588 J

We also know that the kinetic energy of the pendulum bob and the 1 kg mass after the collision is equal to the gravitational potential energy of the pendulum bob at its initial height, which is given by:

Gravitational potential energy = mgh

where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the initial height of the pendulum bob. Substituting the values given in the problem, we get:

Gravitational potential energy = 0.5 * 9.81 * 0.15

= 0.73575 J

Since the collision is perfectly elastic, the kinetic energy of the pendulum bob and the 1 kg mass is also equal to the total initial kinetic energy of the pendulum bob, which is given by:

Initial kinetic energy = (1/2)mv^2

where m is the mass of the pendulum bob, and v is its velocity just before the collision. We can use the principle of conservation of momentum to find the velocity of the pendulum bob just before the collision. Since the momentum is conserved, we have the following:

m1v1 + m2v2 = (m1 + m2)v

Where m1 and v1 are the mass and velocity of the pendulum bob just before the collision, m2 and v2 are the mass and velocity of the 1 kg mass just before the collision, and v is the combined system just after the collision. Since the 1 kg mass is initially at rest, we can simplify this equation to:

m1v1 = (m1 + m2)v

Substituting the given values, we get:

0.5v1 = (0.5 + 1)v

v = 0.25v1

We can now use the principle of conservation of energy to find v1, since the total initial energy is equal to the sum of the gravitational potential energy and the kinetic energy just before the collision. We have:

0.5mv1^2 + mgh = 0.73575 J

Substituting the given values and solving for v1, we get:

v1 = sqrt(2gh/0.5)

= sqrt(2 * 9.81 * 0.15 / 0.5)

= 1.17 m/s

To learn more about principle of conservation, click on the given link.

https://brainly.in/question/1552096

To learn more about energy and momentum, click on the given link.

https://brainly.in/question/239563

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