1.A roller coaster is towed up an incline at a steady speed of 0.500 m/s by a chain parallel to the surface of the incline. The slope is 3.12 %, which means that the elevation increases by 3.12 m for every 100 m of horizontal distance. The mass of the roller coaster is 415 kg. Neglecting friction, what is the magnitude of the force exerted on the roller coaster by the chain?

2.A fire helicopter carries a 563 kg bucket of water at the end of a 15.8 m long cable. Flying back from a fire at a constant speed of 40.4 m/s, the cable makes an angle of 36.0° with respect to the vertical. Determine the force of air resistance on the bucket.
3.If a bicyclist of mass 62.0 kg (including the bicycle) can coast down a 6.50o hill at a steady speed of 5.90 km/hr because of air resistance, how much force must be applied to climb the hill at the same speed (and the same air resistance)?

1. The tangent of the slope angle is 0.0312. That makes the angle A = 1.787 degrees from horizontal.

In a frictionless situation, the chain force F equals the component of the weight force in the opposite direction down the ramp. F = M g sin A

Please refrain from multiple questions per post and name changes.

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1. To find the magnitude of the force exerted on the roller coaster by the chain, we can use the equation: Force = mass * acceleration.

First, we need to find the acceleration of the roller coaster. Since it is being towed up an incline at a steady speed, the net force acting on it must be zero. This means that the force exerted by the chain (in the upward direction) must balance the force of gravity (in the downward direction).

The force of gravity can be calculated using the equation: force of gravity = mass * gravity, where gravity is the acceleration due to gravity (approximately 9.8 m/s^2).

Next, we need to calculate the vertical component of the force of gravity. We can do this by multiplying the force of gravity by the sine of the angle of the incline (3.12%).

Now, the force exerted by the chain must be equal to the vertical component of the force of gravity. So, the magnitude of the force exerted on the roller coaster by the chain is equal to the vertical component of the force of gravity.

2. To determine the force of air resistance on the bucket, we need to use the equation: Force of air resistance = 0.5 * rho * A * Cd * v^2, where rho is the density of air, A is the cross-sectional area of the bucket, Cd is the drag coefficient, and v is the velocity of the bucket.

First, we need to calculate the cross-sectional area of the bucket. As the bucket is perpendicular to the cable, the cross-sectional area will be a circle with a diameter equal to the width of the bucket.

Next, we also need to know the value of the density of air, which is usually given and can be obtained from reference materials.

Finally, we use the given values of the velocity of the bucket (40.4 m/s) and the angle of the cable with respect to the vertical (36.0°) to find the force of air resistance on the bucket using the equation mentioned earlier.

3. To find the force required to climb the hill at the same speed and with the same air resistance, we need to consider the forces acting on the bicyclist.

At a steady speed, the net force is zero. So, the force applied by the bicyclist to climb the hill must balance the force of gravity and the force of air resistance.

The force of gravity can be calculated using the equation: force of gravity = mass * gravity.

The force of air resistance can be determined using the equation mentioned in question 2.

To climb the hill at the same speed, the force applied by the bicyclist must be equal to the sum of the force of gravity and the force of air resistance.