A uniform disk with a mass of 130 kg and a radius of 1.2 m rotates initially with an angular speed of 950 rev/min. A constant tangential force is applied at a radial distance
of 0.5 m.
How much work must this force do to stop the wheel?
Answer in units of kJ
To stop the wheel, the tangential force will have to do an
amount of work equal to the initial rotational kinetic energy of the wheel.
ω =2•π•n = 2• π•950/60 rad/s
W = I•ω^2/2 = m•R^2• ω^2/4,
where ω =2•π•n = 2• π•950/60 rad/s.
The tangential force and the distance d = 0.5 m are unnecessary for calculation.
a rocket initially at rest on the ground lifts off vertically with a constant acceleration of 2.0 X 10^4 meter per second squared. How long will it take the rocket to reach an altitude of 9.0 X 10^3 meter?
To find the work done to stop the wheel, we need to calculate the initial kinetic energy of the wheel.
1. Start by converting the angular speed from rev/min to rad/s:
- 1 revolution is equal to 2π radians.
- So, the angular speed in rad/s is: 950 rev/min * (2π rad/1 revolution) * (1 min/60 s).
- Calculate to find: 950 rev/min * (2π rad/1 revolution) * (1 min/60 s) = 99.32 rad/s.
2. Now, calculate the initial kinetic energy of the wheel using the formula:
- Kinetic energy (K) = (1/2) * I * ω^2,
where I is the moment of inertia and ω is the angular velocity.
3. The moment of inertia for a uniform disk about its axis of rotation is given by:
- I = (1/2) * m * r^2,
where m is the mass of the disk and r is its radius.
4. Plug in the given values into the equation:
- I = (1/2) * 130 kg * (1.2 m)^2
= 93.6 kg·m^2.
5. Calculate the kinetic energy (K):
- K = (1/2) * (93.6 kg·m^2) * (99.32 rad/s)^2
= 462,329.34 J.
6. Finally, the work done by the constant tangential force to stop the wheel will be equal to the initial kinetic energy:
- Work = 462,329.34 J.
7. To express the work in kilojoules, divide by 1000:
- Work = 462,329.34 J / 1000
= 462.33 kJ.
Therefore, the work done by the force to stop the wheel is approximately 462.33 kJ.
To calculate the work done to stop the wheel, we can use the relationship between work and kinetic energy. The work done by a force is equal to the change in kinetic energy.
1. First, let's find the initial angular velocity in rad/s. We know that 1 revolution is equal to 2π radians, so we can convert the initial angular velocity from rev/min to rad/s:
Angular velocity (ω) = (950 rev/min) * (2π rad/rev) * (1 min/60 s)
2. Next, we need to find the initial kinetic energy of the rotating disk. The kinetic energy of a rotating object can be calculated using the formula:
Kinetic energy = (1/2) * I * ω^2
where I is the moment of inertia of the disk. The moment of inertia of a uniform disk is given by:
I = (1/2) * m * r^2
where m is the mass of the disk and r is the radius.
Plugging in the given values:
I = (1/2) * (130 kg) * (1.2 m)^2
Kinetic energy = (1/2) * (1/2) * (130 kg) * (1.2 m)^2 * (ω)^2
3. The work done by the applied force is equal to the change in kinetic energy of the rotating disk. Since the disk is being brought to a stop, the final kinetic energy will be zero, and the work done will be equal to the initial kinetic energy:
Work = (1/2) * (1/2) * (130 kg) * (1.2 m)^2 * (ω)^2
4. Finally, we can convert the work from joules to kilojoules by dividing the result by 1000:
Work = [(1/2) * (1/2) * (130 kg) * (1.2 m)^2 * (ω)^2] / 1000
Calculate the above expression to find the work done to stop the wheel in units of kJ.