A 4.74 kg spherical ball with radius 1.4 cm rolls on a track with a loop of radius 39 cm that sits on a table 1.2 meters above the ground. The last 1 meter section of the track is horzontal and is not connected to the rest of the track. 0.33 meters of this section of the track hangs over the edge of the table. The track has a linear mass density of 0.82 kg/m.
How far does the ball travel along the loose length of track before the track starts to tip?
can you please check my work thanks
Torqueball = Torquetrack
4.75(g)x = 0.82g(0.5)
x = .82/.4.75 (0.5) = 0.086498
it travels 0.33m - 0.089498 =0.24m
The track mass is 0.82 kg and its CG is 0.50 - 0.33 = 0.17 m from the edge of the table. When the ball has travelled x meters past the edge and tipping begins, x*4.74 = 0.82*0.17
x = 2.9*10^-2 m = 2.9 cm
To determine how far the ball travels along the loose length of the track before the track starts to tip, we first need to calculate the torque of the ball and the torque of the track.
The torque of the ball can be calculated using the formula: Torque = mass * acceleration * radius, where mass is the mass of the ball, acceleration is the acceleration due to gravity (9.8 m/s^2), and radius is the radius of the ball.
Torque of the ball = (4.74 kg) * (9.8 m/s^2) * (0.014 m) = 0.66867 Nm
The torque of the track can be calculated using the formula: Torque = linear mass density * gravitational acceleration * radius * angle, where linear mass density is the mass per unit length of the track, gravitational acceleration is the acceleration due to gravity (9.8 m/s^2), radius is the radius of the track, and angle is the angle at which the track starts to tip.
Torque of the track = (0.82 kg/m) * (9.8 m/s^2) * (3.9 m) * (angle)
Setting the torque of the ball equal to the torque of the track and solving for the angle, we have:
0.66867 Nm = (0.82 kg/m) * (9.8 m/s^2) * (3.9 m) * (angle)
angle = 0.66867 Nm / ((0.82 kg/m) * (9.8 m/s^2) * (3.9 m))
angle ≈ 0.021 radians
The length of the horizontal section of the track that hangs over the edge of the table is given as 0.33 m. To calculate the distance the ball travels along the loose length of the track before the track starts to tip, we need to subtract the distance traveled along the inclined section of the track, given by L = (0.021 radians) * (39 cm).
L = (0.021 rad) * (0.39 m) ≈ 0.00819 m
Therefore, the distance the ball travels along the loose length of the track before the track starts to tip is approximately 0.33 m - 0.00819 m ≈ 0.321 m.