A solid sphere with a radius of 0.2 meters and a mass of 5 kg rolls down an incline without slipping. Calculate its acceleration, the time it takes to reach the bottom of the incline, and its final velocity.
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To calculate the acceleration, we can use Newton's second law, which states that the net force acting on an object is equal to the product of its mass and acceleration.
The net force is the component of the gravitational force acting down the incline, which is equal to the mass of the object multiplied by the acceleration due to gravity (9.8 m/s²) multiplied by the sine of the angle of the incline.
The mass of the object is 5 kg.
The angle of the incline is not given, so we'll assume it's a 45-degree incline, which means the sine of the angle is 1/sqrt(2).
The net force can be calculated as follows:
Net Force = mass * acceleration = 5 kg * 9.8 m/s² * (1/sqrt(2))
Now, we can use the net force to calculate the acceleration:
5 kg * 9.8 m/s² * (1/sqrt(2)) = 34.68 N
Net force = mass * acceleration
34.68 N = 5 kg * acceleration
acceleration = 34.68 N / 5 kg = 6.94 m/s²
The time it takes for the sphere to reach the bottom can be calculated using the equation of motion:
vf = vi + at
Where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time.
The sphere starts from rest, so its initial velocity is 0 m/s.
The final velocity is what we need to calculate.
vf = vi + at
vf = 0 m/s + (6.94 m/s²)(t)
vf = 6.94 m/s²t
The sphere will reach the bottom when its final velocity is zero, so we can set vf equal to zero and solve for t:
0 = 6.94 m/s²t
t = 0 s (initially) or when the sphere reaches the bottom
So, at the bottom of the incline, the final velocity of the sphere is zero.
To summarize:
acceleration = 6.94 m/s²
time to reach the bottom = 0 s
final velocity = 0 m/s.
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