A 2.00-kg object is hanging on a 1.1m string. The object is pulled 20cm away from the equilibrium position and held there.
a) What is the pulling force in the string when the object is kept still ?
b) Now the object is released, what is the speed of the object when it goes through the central position ?
Please help :)
a) To calculate the pulling force in the string when the object is kept still, we need to consider the weight of the object and the tension in the string.
The weight of the object is given by the equation: weight = mass x gravitational acceleration
Given that the mass of the object is 2.00 kg and the gravitational acceleration is approximately 9.8 m/s², we can calculate the weight as follows:
weight = 2.00 kg x 9.8 m/s² = 19.6 N
Since the object is held still, the tension in the string must be equal to the weight. Therefore, the pulling force in the string is also 19.6 N.
b) To calculate the speed of the object when it goes through the central position after being released, we can use the principle of conservation of mechanical energy.
The potential energy at the maximum displacement is converted entirely into kinetic energy when the object reaches the central position. Since there is no loss in energy, we can equate the potential energy to the kinetic energy.
The potential energy of the object is given by the equation: potential energy = mass x gravitational acceleration x height.
The height can be calculated as the difference between the length of the string and the displacement from the equilibrium position. Given that the string length is 1.1 m and the displacement is 0.20 m, we have:
height = 1.1 m - 0.20 m = 0.90 m
The potential energy can now be calculated as:
potential energy = 2.00 kg x 9.8 m/s² x 0.90 m = 17.64 J
At the central position, all the potential energy is converted into kinetic energy, which is given by the equation: kinetic energy = (1/2) x mass x velocity².
We can rearrange this equation to solve for velocity:
velocity = sqrt(2 x potential energy / mass)
Substituting the values we know, we have:
velocity = sqrt(2 x 17.64 J / 2.00 kg) = sqrt(17.64 m²/s²) = 4.2 m/s
Therefore, the speed of the object when it goes through the central position is 4.2 m/s.
To find the answers to these questions, we need to apply the principles of simple harmonic motion.
a) To determine the pulling force in the string when the object is kept still, we can use Hooke's Law. According to Hooke's Law, the force exerted by a spring or string is directly proportional to the displacement from equilibrium.
The formula for Hooke's Law is: F = k * x,
where F is the force, k is the spring constant (also known as the stiffness or the force constant), and x is the displacement.
In this case, the string acts like a spring, and the displacement is given as 20 cm (which is equal to 0.20 m). The force we need to find is the pulling force, which is equivalent to the tension in the string.
So, the equation becomes: T = k * x
We need to find the spring constant, k. The formula for the spring constant is:
k = (mass * acceleration due to gravity) / (extension or elongation of the spring)
In this scenario, the extension or elongation of the string is 1.1 m (given as the length of the string). The acceleration due to gravity is approximately 9.8 m/s^2. The mass of the object is 2.0 kg.
Plugging in these values, we can calculate the spring constant, k.
k = (2.0 kg * 9.8 m/s^2) / 1.1 m
Calculating this expression gives us the value of k. Once we have the spring constant, we can find the pulling force, T, by substituting the values back into the equation T = k * x.
b) To determine the speed of the object when it goes through the central position, we can use the principle of conservation of energy.
The total mechanical energy of the system, consisting of potential energy and kinetic energy, remains constant.
At the maximum displacement, when the object is pulled away from the equilibrium position, it possesses maximum potential energy and minimum kinetic energy. At the central position, the object has maximum kinetic energy and minimum potential energy.
The potential energy at the maximum displacement is given by: PE = (1/2) * k * x^2
Where PE is the potential energy, k is the spring constant, and x is the displacement.
The kinetic energy at the central position is given by: KE = (1/2) * m * v^2
Where KE is the kinetic energy, m is the mass of the object, and v is the velocity.
Since total mechanical energy is conserved, we can equate the potential energy at maximum displacement to the kinetic energy at the central position:
(1/2) * k * x^2 = (1/2) * m * v^2
We can rearrange this equation to solve for the velocity, v:
v = sqrt((k * x^2) / m)
By substituting the known values for k, x, and m, you can calculate the speed of the object when it goes through the central position.