A steel ball bearing the radius 1 cm is rolling along a table with speed 20 cm/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 the steel ball bearing will rise before stopping, we can use the principle of conservation of mechanical energy. The initial kinetic energy (KE) of the ball bearing is converted into potential energy (PE) as it rolls up the incline.
First, let's calculate the initial kinetic energy of the ball bearing:
Kinetic Energy (KE) = 0.5 * mass * velocity^2
The mass of the ball bearing can be calculated using the formula for the volume of a sphere:
Volume = (4/3) * π * radius^3
Assuming the density of steel is constant, we can calculate the mass:
Mass = density * Volume
Next, we can calculate the potential energy at the highest point of the incline using the formula:
Potential Energy (PE) = mass * gravity * height
Since we want to find the height the ball bearing reaches before stopping, we can set the potential energy equal to the initial kinetic energy:
PE = KE
mass * gravity * height = 0.5 * mass * velocity^2
Canceling out the mass from both sides of the equation:
gravity * height = 0.5 * velocity^2
Now we can solve for the height:
height = (0.5 * velocity^2) / gravity
Plugging in the given values:
height = (0.5 * (20 cm/sec)^2) / (9.8 m/s^2)
Note that the units for velocity and acceleration need to be consistent, so we convert cm/sec to m/s:
height = (0.5 * (0.2 m/s)^2) / (9.8 m/s^2)
Simplifying:
height = 0.00204 m
Therefore, the ball bearing will rise approximately 0.00204 meters (or 2.04 millimeters) above the table level before stopping, assuming no friction losses.