A ball with a mass of 2kg rests on an incline with an angle of 30degrees .The ball is to be released to roll down the incline plane, neglecting all friction. Determine :
A. The loss in potential energy after it has rolled 14metres
B. The velocity after it has rolled 14metres
A. it has dropped 14sin30° = 7m, so PE lost is mgh = 2*9.8*7 = ____ J
B. KE gained = PE lost, 1/2 mv^2 = PE lost. solve for v
A. As the ball rolls down the incline plane, it loses potential energy. The potential energy lost can be calculated using the formula:
Potential Energy lost = mass * gravity * height
Here, the height is the vertical distance the ball rolls down, which can be calculated using the angle of the incline and the distance it rolls. The height can be found using the formula:
Height = distance * sin(angle)
Given that the mass of the ball is 2kg, gravity is approximately 9.8 m/s^2, and the angle is 30 degrees, we can substitute the values into the formulas:
Height = 14m * sin(30 degrees) = 7m * 0.5 = 3.5m
Potential Energy lost = 2kg * 9.8 m/s^2 * 3.5m = 68.6 Joules
Therefore, the ball loses 68.6 Joules of potential energy after it has rolled 14 meters.
B. To determine the velocity of the ball after it has rolled 14 meters, we can use the conservation of energy. The potential energy lost is converted into kinetic energy. The formula for kinetic energy is:
Kinetic Energy = 0.5 * mass * velocity^2
Equating the potential energy lost to the kinetic energy gained:
Potential Energy lost = Kinetic Energy gained
68.6 Joules = 0.5 * 2kg * velocity^2
Simplifying the equation:
velocity^2 = 68.6 Joules / (0.5 * 2kg) = 68.6 Joules / 1kg = 68.6 m^2/s^2
velocity = sqrt(68.6 m^2/s^2) = 8.28 m/s
Therefore, the velocity of the ball after it has rolled 14 meters is approximately 8.28 m/s.
To solve this problem, we need to use the principles of conservation of mechanical energy.
A. Loss in Potential Energy:
The potential energy of an object can be calculated using the formula:
Potential Energy (PE) = mass (m) * gravity (g) * height (h)
Given:
Mass (m) = 2 kg
Height (h) = 14 m
First, we need to find the vertical height component of the incline (h') using trigonometry.
h' = h * sin(theta)
h' = 14 m * sin(30)
h' = 14 m * 0.5
h' = 7 m
Now, we can calculate the initial potential energy (PE_initial) and the final potential energy (PE_final).
PE_initial = m * g * h
PE_final = m * g * h'
Since the ball is at rest initially, all of its potential energy can be converted into kinetic energy while rolling down the inclined plane.
PE_initial = Kinetic Energy (KE) + PE_final
Loss in Potential Energy = PE_initial - PE_final
Let's calculate the loss in potential energy:
PE_initial = 2 kg * 9.8 m/s^2 * 14 m
PE_final = 2 kg * 9.8 m/s^2 * 7 m
Loss in Potential Energy = PE_initial - PE_final
B. Velocity after rolling 14 meters:
Since the ball starts from rest and we neglected friction, all its initial potential energy is converted into kinetic energy.
KE = PE_initial - Loss in Potential Energy
The formula for kinetic energy is given by:
KE = (1/2) * mass * velocity^2
Substituting the values, we can solve for velocity (v):
KE = (1/2) * 2 kg * v^2
PE_initial - Loss in Potential Energy = (1/2) * 2 kg * v^2
Now, solve for v:
v = sqrt((PE_initial - Loss in Potential Energy) / (mass * 0.5))
Let's calculate the velocity:
v = sqrt((PE_initial - Loss in Potential Energy) / (2 kg * 0.5))
Now, you can use the given values to calculate the loss in potential energy and the velocity.
To solve this problem, we can use the principles of conservation of energy. We can consider the potential energy of the ball at its initial position and its kinetic energy at its final position.
First, let's calculate the loss in potential energy (PE) after it has rolled 14 meters:
The formula for potential energy is PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity (~9.8 m/s^2), and h is the vertical height.
In this case, the height (h) is given by the vertical distance traveled by the ball along the incline, which is h = 14 m * sin(30°).
Now, substituting the given values into the formula, we have:
PE = 2 kg * 9.8 m/s^2 * 14 m * sin(30°)
Simplifying the equation:
PE = 2 kg * 9.8 m/s^2 * 14 m * 0.5
PE = 137.2 J
Therefore, the loss in potential energy after it has rolled 14 meters is 137.2 Joules.
Next, let's determine the velocity of the ball after it has rolled 14 meters:
The formula for kinetic energy is KE = 0.5mv^2, where KE is the kinetic energy, m is the mass of the ball, and v is the velocity.
Since the ball is rolling without friction, all of the potential energy goes into kinetic energy:
KE = PE = 137.2 J
Now, we can solve for the velocity:
137.2 J = 0.5 * 2 kg * v^2
137.2 J = v^2
Taking the square root of both sides:
v ≈ sqrt(137.2 J) ≈ 11.7 m/s
Therefore, the velocity of the ball after it has rolled 14 meters is approximately 11.7 m/s.