Minimum force required to prevent a ball weighing 29.6 pounds from rolling down a ramp which is inclined 14.9 degrees with the horizon?
To calculate the minimum force required to prevent a ball from rolling down a ramp, we need to consider the force of gravity acting on the ball and the component of this force that is parallel to the ramp.
First, let's calculate the force of gravity acting on the ball. The force of gravity is given by the formula:
F_gravity = m * g
Where:
m = mass of the ball
g = acceleration due to gravity (approximately 32.2 ft/s^2)
Given that the weight of the ball is 29.6 pounds, we need to convert it to mass in terms of pounds. We know that 1 pound is equal to 0.4536 kilograms. Therefore:
m = 29.6 lb * (0.4536 kg / 1 lb) = 13.4164 kg
Next, we can calculate the force of gravity:
F_gravity = 13.4164 kg * 32.2 ft/s^2 = 431.4648 lb⋅ft/s^2
Now, let's find the component of the force of gravity that acts parallel to the ramp. This component is given by the formula:
F_parallel = F_gravity * sin(theta)
Where:
theta = angle of the ramp with respect to the horizontal (14.9 degrees)
Converting the angle from degrees to radians:
theta_radians = 14.9 degrees * (pi/180 degrees) = 0.2590 radians
Now, let's calculate the component of the force of gravity parallel to the ramp:
F_parallel = 431.4648 lb⋅ft/s^2 * sin(0.2590) = 431.4648 lb⋅ft/s^2 * 0.2499 = 107.8470 lb⋅ft/s^2
Therefore, the minimum force required to prevent the ball from rolling down the ramp is approximately 107.8470 pounds of force.
To determine the minimum force required to prevent the ball from rolling down the ramp, we need to consider the forces acting on the ball.
The most significant force to consider is the force of gravity acting on the ball, which can be calculated using the weight of the ball. The weight is the force exerted on an object due to gravity and can be calculated using the formula:
Weight = mass * acceleration due to gravity
Given that the weight of the ball is 29.6 pounds, we can use the fact that 1 pound is approximately equal to 0.454 kg to convert the weight into the mass of the ball:
Mass = Weight / acceleration due to gravity
Now, the force of gravity acting on the ball can be calculated using the mass of the ball and the acceleration due to gravity, which is approximately 9.8 m/s²:
Force of gravity = Mass * acceleration due to gravity
Next, to prevent the ball from rolling down the inclined ramp, we need to apply a force in the opposite direction of the force of gravity. This force is known as the normal force and is equal in magnitude to the component of the weight acting perpendicular to the ramp.
The normal force can be calculated using the formula:
Normal force = Weight * cos(angle of inclination)
Given that the angle of inclination is 14.9 degrees, we need to convert it into radians to use trigonometric functions. The conversion is as follows:
Angle in radians = Angle in degrees * π / 180
Finally, the minimum force required to prevent the ball from rolling down the ramp can be calculated by adding the force of gravity and the force needed to counteract the component of the weight along the ramp.
Minimum force = Force of gravity + Weight * sin(angle of inclination)
Substituting the calculated values into the formula will give you the minimum force required to prevent the ball from rolling down the ramp.