Problem 72 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A 75 -kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a \(23^{\circ}\) hill. The skier is pulled a distance \(x=220 \mathrm{~m}\) along the incline and it takes 2.0 min to reach the top of the hill. If the coefficient of kinetic friction between the snow and skis is \(\mu_{\mathrm{k}}=0.10,\) what horsepower engine is required if 30 such skiers (max) are on the rope at one time?

Step-by-Step Solution

Answer

The engine requires approximately 26.2 horsepower.

Step 1: Calculate the forces involved on one skier

First, we need to find the forces acting on one skier on the inclined plane. The gravitational force causing the skier to slide down is given by \( F_{\text{gravity}} = mg\sin(\theta) \). With \( m = 75 \) kg, \( g = 9.8 \text{ m/s}^2 \), and \( \theta = 23^\circ \), we have:\[ F_{\text{gravity}} = 75 \times 9.8 \times \sin(23^\circ) \approx 287.6 \text{ N} \]Next, calculate the force of friction, which is \( F_{\text{friction}} = \mu_k mg\cos(\theta) \):\[ F_{\text{friction}} = 0.10 \times 75 \times 9.8 \times \cos(23^\circ) \approx 67.9 \text{ N} \]

Step 2: Determine the net force and work done for one skier

Since the skier is moving at a constant speed, the net force \( F_{\text{net}} \) is zero, meaning that the tension in the rope \( T \) must equal the sum of the gravitational component pulling down the slope and the frictional force, hence:\[ T = F_{\text{gravity}} + F_{\text{friction}} = 287.6 + 67.9 = 355.5 \text{ N} \]The work done on one skier \( W \) is given by the formula \( W = T \times x \). With \( x = 220 \text{ m} \), we have:\[ W = 355.5 \times 220 \approx 78,210 \text{ J} \]

Step 3: Compute the power required for one skier

The power \( P \) is the work done per unit time. It takes 2 minutes, or 120 seconds, for the skier to reach the top:\[ P = \frac{W}{t} = \frac{78,210}{120} \approx 651.75 \text{ W} \]

Step 4: Calculate the total power for 30 skiers

Since the power calculated above is for one skier, we need to multiply the power by the number of skiers, which is 30:\[ P_{\text{total}} = 30 \times 651.75 \approx 19,552.5 \text{ W} \]

Step 5: Convert the total power into horsepower

Horsepower is a unit of power where 1 horsepower is equivalent to 746 watts. Thus, convert the total power from watts to horsepower:\[ P_{\text{hp}} = \frac{19,552.5}{746} \approx 26.2 \text{ hp} \]

Step 6: Conclusion

The engine must have a power output of approximately 26.2 horsepower to pull 30 skiers up the hill at the same time.

Key Concepts

Inclined Plane

An inclined plane is a flat surface tilted at an angle to the horizontal. It's commonly used to move objects up or down distances with less effort than lifting or lowering directly. The angle of the plane, in this case, is 23 degrees, which influences several factors in physics calculations, including how forces break down along the axes.

When dealing with inclined planes, it's crucial to understand how gravity affects movement. On an incline, the force of gravity can be split into two components:

• A component parallel to the plane, which pulls the object down the slope.
• A component perpendicular to the plane, which presses the object into the surface.

The parallel component is calculated with the formula \( F_{\text{gravity, parallel}} = mg \sin(\theta) \). Here, \( m \) is mass, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of incline. This parallel component is what the skier is countering with the engine-powered rope.

Gravitational Force

Gravitational force is the attractive force between two masses. For objects near Earth's surface, this force can be calculated as the object’s weight, which equals mass times the gravitational acceleration \( g = 9.8 \text{ m/s}^2 \).

On an inclined plane, gravity has a different effect because the surface alters the direction of force interactions:

• Weight no longer acts directly downwards relative to the surface.
• The gravitational force’s horizontal component affects motion along the incline.
• The vertical component of gravitational force presses the object into the inclined surface, influencing friction.

Combining gravitational force with friction gives the net force acting along the slope, expressed as the sum of gravitational and frictional forces. In this example, the skier must overcome both of these to move uphill, indicating how each force influences total work.

Power Calculation

Power is the rate at which work is done. It’s an essential physics concept, often discussed regarding engines and humans performing tasks.

For this skier scenario, power calculation starts with first finding the work done, which relies on force and distance. Work \( W \) is calculated by multiplying the tension in the rope by the distance moved along the incline, \( W = T \times x \).

Once we know work, power \( P \) is found by dividing work by time, \( P = \frac{W}{t} \). This gives power in watts (W), which can then be converted to horsepower (1 horsepower = 746 watts) for applications like engines:

• Calculate total energy for movement: \( W = 78,210 \text{ J} \)
• Determine the power needed over a certain duration: \( P = 651.75 \text{ W} \)
• Convert watts into horsepower to understand engine requirements: \( P_{\text{hp}} = 26.2 \text{ hp} \)

This process illustrates how energy, force, and time unite to define power, helping to determine the right engine needed for pulling multiple skiers.