A street ball bearing of radius 0.50 cm is rolling along a table at 20cm/sec when it starts to roll up an incline. How high above the table level will it rise before stopping? Ignore friction losses.
To determine how high above the table level the ball will rise, we need to consider the energy conservation principle.
The initial energy of the ball consists of its kinetic energy as it is rolling on the table, given by:
KE_initial = (1/2) * m * v^2,
where m is the mass of the ball and v is its initial velocity.
However, since we are given the radius of the ball instead of its mass, we need to convert the radius to mass. Assuming the ball is made of a uniform material with a known density, we can calculate the mass using the formula:
m = density * volume,
where volume = (4/3) * π * r^3.
Given that the ball bearing has a radius of 0.50 cm (0.005 m) and assuming a density of steel (which is commonly used for ball bearings) as 7850 kg/m^3, we can calculate the mass of the ball:
volume = (4/3) * π * (0.005)^3 ≈ 5.24 x 10^-8 m^3,
m = 7850 kg/m^3 * 5.24 x 10^-8 m^3 ≈ 0.41 g ≈ 0.00041 kg.
Now, we can substitute the mass and initial velocity (given as 20 cm/sec, or 0.20 m/sec) into the equation for kinetic energy to find the initial energy:
KE_initial = (1/2) * 0.00041 kg * (0.20 m/sec)^2 ≈ 8.2 x 10^-6 J.
As the ball starts rolling up the incline, it loses kinetic energy and gains potential energy according to the principle of energy conservation. When the ball stops rolling, all of its initial kinetic energy will be converted into potential energy. Therefore, we can equate the initial kinetic energy to the potential energy gained when the ball reaches its maximum height, h:
PE_final = m * g * h,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the known values, we have:
8.2 x 10^-6 J = 0.00041 kg * 9.8 m/s^2 * h.
Simplifying the equation, we can solve for h:
h = (8.2 x 10^-6 J) / [(0.00041 kg) * (9.8 m/s^2)]
≈ 2 m.
Therefore, the ball will rise approximately 2 meters above the level of the table before stopping.