Consider a solid ball of mass m = 50g and is placed on a long flat slope that makes an angle of 30° to the horizontal. The ball was initially at rest and was then released such that it rolled down the slope without slipping. With a detailed explanation of your method determine the speed of the ball after it had rolled 4.0m down the slope. You may assume acceleration due to gravity is 10.0ms^-1 and that the moment of inertia of a solid sphere is given by the formula l=2/5mr^2.
To determine the speed of the ball after it has rolled 4.0m down the slope, we first need to calculate the potential energy the ball gains when it is at the top of the slope, and then convert it into kinetic energy.
1. Calculate the potential energy:
The potential energy of an object is given by the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object.
In this case, the height h is the vertical distance between the starting point of the ball and the point where it stops after rolling down 4.0m.
To find this height, we can use trigonometry:
h = (4.0m) * sin(30°) (since sin(30°) = 0.5)
h = 2.0m
Therefore, the potential energy at the top of the slope is:
PE = (0.05kg) * (10.0m/s^2) * (2.0m)
PE = 1.0 J
2. Convert potential energy to kinetic energy:
The potential energy gained by the ball is converted entirely into kinetic energy when it reaches the bottom of the slope, assuming there is no loss of energy due to friction or other factors.
The kinetic energy is given by the formula KE = 0.5mv^2, where m is the mass and v is the velocity (speed) of the ball.
Since the potential energy gained is equal to the kinetic energy, we have:
1.0 J = 0.5 * (0.05kg) * v^2
Simplifying the equation:
v^2 = (2 * 1.0 J) / (0.05kg)
v^2 = 40 J/kg
Taking the square root of both sides:
v = sqrt(40 J/kg)
v = 6.32 m/s
Therefore, the speed of the ball after it has rolled 4.0m down the slope is approximately 6.32 m/s.