What is the kinetic energy of a 150gram object moving at a velocity of 100m/s?
1 answer:
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Answer:
a) v = 1.01 m/s
b) a = 5.6 m/s²
Explanation:
a)
- If the disk is initially at rest, and it is applied a constant force tangential to the rim, we can apply the following expression (that resembles Newton's 2nd law, applying to rigid bodies instead of point masses) as follows:
- Where τ is the external torque applied to the body, I is the rotational inertia of the body regarding the axis of rotation, and α is the angular acceleration as a consequence of the torque.
- Since the force is applied tangentially to the rim of the disk, it's perpendicular to the radius, so the torque can be calculated simply as follows:
- τ = F*r (2)
- For a solid uniform disk, the rotational inertia regarding an axle passing through its center is just I = m*r²/2 (3).
- Replacing (2) and (3) in (1), we can solve for α, as follows:
- Since the angular acceleration is constant, we can use the following kinematic equation:
- Prior to solve it, we need to convert the angle rotated from revs to radians, as follows:
- Replacing (6) in (5), taking into account that ω₀ = 0 (due to the disk starts from rest), we can solve for ωf, as follows:
- Now, we know that there exists a fixed relationship the tangential speed and the angular speed, as follows:
- where r is the radius of the circular movement. If we want to know the tangential speed of a point located on the rim of the disk, r becomes the radius of the disk, 0.200 m.
- Replacing this value and (7) in (8), we get:
b)
- There exists a fixed relationship between the tangential and the angular acceleration in a circular movement, as follows:
- where r is the radius of the circular movement. In this case the point is located on the rim of the disk, so r becomes the radius of the disk.
- Replacing this value and (4), in (9), we get:
- Now, the resultant acceleration of a point of the rim, in magnitude, is the vector sum of the tangential acceleration and the radial acceleration.
- The radial acceleration is just the centripetal acceleration, that can be expressed as follows:
- Since we are asked to get the acceleration after the disk has rotated 0.2 rev, and we have just got the value of the angular speed after rotating this same angle, we can replace (7) in (11).
- Since the point is located on the rim of the disk, r becomes simply the radius of the disk,, 0.200 m.
- Replacing this value and (7) in (11) we get:
- The magnitude of the resultant acceleration will be simply the vector sum of the tangential and the radial acceleration.
- Since both are perpendicular each other, we can find the resultant acceleration applying the Pythagorean Theorem to both perpendicular components, as follows:
1) Magnitude of the force:
The magnetic field generated by a current-carrying wire is
where
is the vacuum permeability
I is the current in the wire
r is the distance at which the field is calculated
Using I=135 A, the current flowing in each wire, we can calculate the magnetic field generated by each wire at distance
,
which is the distance at which the other wire is located:
Then we can calculate the magnitude of the force exerted on each wire by this magnetic field, which is given by:
2) direction of the force:
The two currents run in opposite direction: this means that the force between them is repulsive. This can be determined by using the right hand rule. Let's apply it to one of the two wires, assuming they are in the horizontal plane, and assuming that the current in the wire on the right is directed northwards:
- the magnetic field produced by the wire on the left at the location of the wire on the right is directed upward (the thumb of the right hand is directed as the current, due south, and the other fingers give the direction of the magnetic field, upward)
Now let's apply the right-hand rule to the wire on the right:
- index finger: current --> northward
- middle finger: magnetic field --> upward
- thumb: force --> due east --> so the force is repulsive
A similar procedure can be used on the wire on the left, finding that the force exerted on it is directed westwards, so the force between the two wires is repulsive.
The magnetic field generated by a current-carrying wire is
where
I is the current in the wire
r is the distance at which the field is calculated
Using I=135 A, the current flowing in each wire, we can calculate the magnetic field generated by each wire at distance
which is the distance at which the other wire is located:
Then we can calculate the magnitude of the force exerted on each wire by this magnetic field, which is given by:
2) direction of the force:
The two currents run in opposite direction: this means that the force between them is repulsive. This can be determined by using the right hand rule. Let's apply it to one of the two wires, assuming they are in the horizontal plane, and assuming that the current in the wire on the right is directed northwards:
- the magnetic field produced by the wire on the left at the location of the wire on the right is directed upward (the thumb of the right hand is directed as the current, due south, and the other fingers give the direction of the magnetic field, upward)
Now let's apply the right-hand rule to the wire on the right:
- index finger: current --> northward
- middle finger: magnetic field --> upward
- thumb: force --> due east --> so the force is repulsive
A similar procedure can be used on the wire on the left, finding that the force exerted on it is directed westwards, so the force between the two wires is repulsive.