a pendilum consists ofa small object hanging from the ceiling at the end of the string of negligible mass. the string has length of 0.7m. with the string hanging vertically,the object is given an initial velocity of 2.0m/s parallel to the ground and swings upwards on a circular arc. eventually,the object comes to a momentary halt at a point where the string makes an a angle with its initial velocity orientation and then swings back downward. find the angle.
To find the angle at which the object momentarily comes to a halt, we can apply the conservation of energy in the system.
Initially, the object has a potential energy (PE) due to its height, and a kinetic energy (KE) due to its velocity.
At the highest point, the object comes to a momentary halt, so its kinetic energy becomes zero.
At this point, all the initial kinetic energy of the object is converted into potential energy.
The potential energy at the highest point (PE) is given by the formula:
PE = m * g * h
where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s^2), and h is the height at the highest point.
In this case, the height is given by the length of the string - h = 0.7 m.
Since the object comes to a momentary halt, the kinetic energy (KE) can be written as:
KE = 0.5 * m * v^2
where v is the initial velocity of the object (2.0 m/s).
Now, equating the potential energy (PE) and kinetic energy (KE), we have:
m * g * h = 0.5 * m * v^2
Canceling out the mass (m) on both sides, we get:
g * h = 0.5 * v^2
Simplifying further, we have:
h = (0.5 * v^2) / g
Plugging in the values, we have:
h = (0.5 * (2.0 m/s)^2) / 9.8 m/s^2
h = (0.5 * 4 m^2/s^2) / 9.8 m/s^2
h = 2 m^2/s^2 / 9.8 m/s^2
h ≈ 0.204 m
Now, to find the angle (θ) at which the string makes with its initial velocity orientation, we can use trigonometry.
The sine of the angle (θ) is given by:
sin(θ) = h / length of string
sin(θ) = 0.204 m / 0.7 m
sin(θ) ≈ 0.2914
To find the angle (θ), we take the inverse sine (arcsin) of the value:
θ ≈ arcsin(0.2914)
θ ≈ 16.9 degrees
Therefore, the object momentarily comes to a halt at a point where the string makes an angle of approximately 16.9 degrees with its initial velocity orientation.
To find the angle at which the object momentarily comes to a halt, we can use the concept of conservation of energy.
The energy of the pendulum is conserved, meaning that the sum of its kinetic and potential energy remains constant throughout its motion.
Initially, when the object is given an initial velocity of 2.0 m/s parallel to the ground, it only has kinetic energy. As it swings upwards, this kinetic energy is converted into potential energy, until it momentarily comes to a halt.
At this moment, the potential energy is at its maximum, and the kinetic energy is zero.
To calculate the angle, we need to equate the initial kinetic energy to the potential energy at the highest point of the swing.
The kinetic energy of the object can be calculated using the formula:
KE = (1/2) * m * v^2,
where m is the mass of the object, and v is its velocity.
The potential energy can be calculated using the formula:
PE = m * g * h,
where g is the acceleration due to gravity, and h is the height.
Since the height is the maximum at the highest point, which is at the angle we want to find, the potential energy at this point is equal to the total mechanical energy of the system.
Therefore, we have:
KE(initial) = PE(at highest point)
Now let's calculate the angle:
1. Determine the mass of the object. If the mass is not given, we cannot determine the angle based on the given information.
2. Calculate the initial kinetic energy:
KE = (1/2) * m * v^2
KE = (1/2) * m * (2.0 m/s)^2
3. Calculate the potential energy at the highest point:
PE = KE(initial)
4. Using the potential energy formula, calculate the height:
PE = m * g * h
h = PE / (m * g)
5. From the given information, we know that the string's length is 0.7m. Using this length and the height calculated in step 4, we can determine the angle using trigonometry.
tan(angle) = height / string length
angle = atan(height / string length)
By following these steps and plugging in the values, you will be able to calculate the angle at which the object momentarily comes to a halt.