A 1-kg mass hangs by a string from a disk with radius 4.6 cm which has a rotational inertia of 5 × 10-5 kg·m2. After it falls a distance of 0.8 meters, how fast is it going to the nearest hundredth of a m/s?
To determine the speed of the 1-kg mass, we can use the principle of conservation of energy. The potential energy at the initial position is equal to the kinetic energy at the final position.
Step 1: Determine the potential energy at the initial position:
The potential energy is equal to the product of the mass (m), acceleration due to gravity (g), and the height (h).
Potential Energy = m * g * h
Given:
Mass (m) = 1 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Height (h) = 0.8 m
Potential Energy = 1 kg * 9.8 m/s^2 * 0.8 m
Potential Energy = 7.84 Joules
Step 2: Determine the kinetic energy at the final position:
The kinetic energy is equal to half the product of the mass and square of the velocity.
Kinetic Energy = (1/2) * m * v^2
Given:
Mass (m) = 1 kg
We need to find the velocity (v). To do that, we need to calculate the change in potential energy and equate it to the kinetic energy.
Change in Potential Energy = Potential Energy - 0 (since height at the final position is 0)
Change in Potential Energy = 7.84 Joules
Equating the change in potential energy to the kinetic energy:
7.84 J = (1/2) * 1 kg * v^2
Step 3: Solve for the velocity (v):
Using the equation above, we can rearrange it to solve for v:
v^2 = (2 * Change in Potential Energy) / m
v^2 = (2 * 7.84 J) / 1 kg
v^2 = 15.68 m^2/s^2
Taking the square root of both sides to find v:
v = √(15.68 m^2/s^2)
v ≈ 3.96 m/s
Therefore, the speed of the 1-kg mass after falling a distance of 0.8 meters is approximately 3.96 m/s to the nearest hundredth of a m/s.
To find the speed of the mass after it falls a distance of 0.8 meters, we can use the principle of conservation of mechanical energy. The potential energy of the mass when it is at a height of 0.8 meters is converted into kinetic energy as it falls.
First, let's determine the potential energy (PE) of the mass when it is at a height of 0.8 meters. The potential energy of an object near the Earth's surface is given by the equation:
PE = m * g * h
where m is the mass (1 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (0.8 meters).
PE = 1 kg * 9.8 m/s^2 * 0.8 m
= 7.84 Joules
Next, the initial potential energy is converted into kinetic energy (KE) as the mass falls. The kinetic energy of an object is given by the equation:
KE = 0.5 * I * ω^2
where I is the rotational inertia (5 × 10^-5 kg·m^2) and ω is the angular velocity. Since the mass is initially at rest, its initial angular velocity is 0.
KE = 0.5 * 5 × 10^-5 kg·m^2 * 0^2
= 0 Joules
According to the conservation of mechanical energy, the initial potential energy will be equal to the final kinetic energy:
PE = KE
7.84 Joules = 0.5 * 5 × 10^-5 kg·m^2 * ω^2
Solving for ω:
ω^2 = (7.84 Joules) / (0.5 * 5 × 10^-5 kg·m^2)
ω^2 = 313600
Taking the square root of both sides:
ω = √313600
ω ≈ 560 radians/s
Finally, to find the linear velocity (v) of the rotating disk, we can use the equation:
v = ω * r
where r is the radius of the disk (4.6 cm).
Converting the radius to meters:
r = 4.6 cm = 0.046 m
v = 560 radians/s * 0.046 m
v ≈ 25.76 m/s
Therefore, the speed of the mass after it falls a distance of 0.8 meters is approximately 25.76 m/s to the nearest hundredth.
change In PE=change in KE
mg*.8=1/2 m v^2 + 1/2 I w^2
but w=v/r
so w^2=v^2/r^2 and I= 1/2 m r^2
so
mg*.8=1/2 m v^2 + 1/4 mv^2
solve for v. Notice the mass m divides out.