The 56 kg man in the roller coaster car is sitting on a bathroom scale. If he is traveling at 40.9 m/s at the point shown and the radius of the vertical coaster track is 87 meters, to the nearest newton what does the scale read? What is the answer to the previous problem if the roller coaster is at the bottom of the track?
I don't know where to even begin for this problem. The only equation I can think of is a(c) = v^2/r but I don't know how to use it/how to apply it to this problem. Please help! Thank you!
at the top: force on scale=mg -mv^2/r
at the bottom: forceonscal=mg+mv^2/r
To solve this problem, you can use the equation for centripetal acceleration:
a(c) = v^2 / r
Given:
Mass of the man (m) = 56 kg
Speed (v) = 40.9 m/s
Radius of track (r) = 87 m
1. Calculate the centripetal acceleration (a(c)) of the roller coaster car:
a(c) = v^2 / r
a(c) = (40.9 m/s)^2 / 87 m
a(c) = 1667.81 m^2/s^2 / 87 m
a(c) ≈ 19.14 m/s^2
2. Calculate the force experienced by the man using Newton's second law of motion:
F = m * a(c)
F = 56 kg * 19.14 m/s^2
F ≈ 1072.64 N
Therefore, the scale would read approximately 1073 Newtons.
If the roller coaster is at the bottom of the track, the situation would be slightly different because there would be an additional force due to gravity acting on the man.
3. Calculate the net force at the bottom of the track:
Net force = F(g) + F(c)
F(g) = m * g (where g is the acceleration due to gravity, approximately 9.8 m/s^2)
F(c) = m * a(c) (centripetal force)
F(g) = 56 kg * 9.8 m/s^2
F(g) = 548.8 N
F(c) = 56 kg * 19.14 m/s^2
F(c) ≈ 1072.64 N
Net force = 548.8 N + 1072.64 N
Net force ≈ 1621.44 N
Therefore, the scale would read approximately 1621 Newtons if the roller coaster is at the bottom of the track.
To solve this problem, we can start by analyzing the forces acting on the man in the roller coaster car. At the given point on the track, there are two forces acting on the man: his weight (mg) and the normal force from the bathroom scale.
Let's break down the problem into two parts:
1. When the roller coaster is at the point shown:
To find the normal force acting on the man, we need to consider the net force acting in the vertical direction. At this point, the net force is the centripetal force required to keep the man moving in a circular path.
The equation we can use is:
Net Force = Centripetal Force,
m * a(c) = m * v^2 / r,
where:
m = mass of the man (56 kg),
a(c) = acceleration (centripetal acceleration),
v = velocity (40.9 m/s),
r = radius of the coaster track (87 m).
First, let's calculate the centripetal acceleration:
a(c) = v^2 / r,
a(c) = (40.9 m/s)^2 / 87 m,
a(c) = 19.0794 m/s^2.
Now, we can determine the net force required:
Net Force = m * a(c),
Net Force = 56 kg * 19.0794 m/s^2,
Net Force = 1069.2864 N.
Since the bathroom scale measures the normal force exerted by the man, the scale will read the same value as the net force acting on the man: 1069.2864 N (rounded to the nearest newton).
2. When the roller coaster is at the bottom of the track:
At the bottom of the track, the man experiences an additional downward force due to gravity since he is upside down. Therefore, the net force acting on the man will be the sum of the centripetal force and his weight.
The equation we can use is:
Net Force = Weight + Centripetal Force,
m * a(c) = m * g + m * v^2 / r,
where:
m = mass of the man (56 kg),
a(c) = acceleration (centripetal acceleration),
g = acceleration due to gravity (9.8 m/s^2),
v = velocity (40.9 m/s),
r = radius of the coaster track (87 m).
First, let's calculate the centripetal acceleration:
a(c) = v^2 / r,
a(c) = (40.9 m/s)^2 / 87 m,
a(c) = 19.0794 m/s^2.
Now, we can determine the net force required:
Net Force = Weight + m * a(c),
Net Force = m * g + m * a(c),
Net Force = 56 kg * 9.8 m/s^2 + 56 kg * 19.0794 m/s^2,
Net Force = 1070.2064 N.
Again, since the bathroom scale measures the normal force exerted by the man, the scale will read the same value as the net force acting on the man: 1070.2064 N (rounded to the nearest newton).
Therefore, the scale reading is approximately 1069.2864 N when the roller coaster is at the point shown, and it is approximately 1070.2064 N when the roller coaster is at the bottom of the track.