Problem 15 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Piles of snow on slippery roofs can become dangerous projectiles as they melt. Consider a chunk of snow at the ridge of a roof with a slope of \(34^{\circ} .(a)\) What is the minimum value of the coefficient of static friction that will keep the snow from sliding down? \((b)\) As the snow begins to melt the coefficient of static friction decreases and the snow finally slips. Assuming that the distance from the chunk to the edge of the roof is \(6.0 \mathrm{~m}\) and the coefficient of kinetic friction is \(0.20,\) calculate the speed of the snow chunk when it slides off the roof. \((c)\) If the edge of the roof is \(10.0 \mathrm{~m}\) above ground, estimate the speed of the snow when it hits the ground.

Step-by-Step Solution

Answer

(a) \( \mu_s \approx 0.67 \), (b) \( v \approx 5.95 \text{ m/s} \), (c) \( v_f \approx 15.77 \text{ m/s} \).

Step 1: Understand the Forces Involved

To solve part (a), we need to consider the forces acting on the snow chunk. The weight of the snow chunk can be divided into components parallel and perpendicular to the roof. The parallel component is responsible for sliding, and the static friction resists this motion.

Step 2: Calculate the Minimum Coefficient of Static Friction

The snow will not slide as long as the static friction force is equal to or greater than the component of the weight parallel to the roof's slope. The frictional force can be expressed as \( f = \mu_s N \), where \( \mu_s \) is the static friction coefficient and \( N \) is the normal force. The parallel component of the gravitational force can be expressed as \( mg \sin(34^\circ) \), and the normal force can be expressed as \( N = mg \cos(34^\circ) \). Thus, the minimum static friction coefficient is \( \mu_s = \tan(34^\circ) \approx 0.67 \).

Step 3: Determine the Kinetic Energy at the Edge of the Roof

For part (b), once the snow starts sliding, kinetic friction comes into play. Using the work-energy principle, the work done by kinetic friction and gravity will equal the change in kinetic energy. The work done by friction is \( W_f = \mu_k mg \cos(34^\circ) \cdot d \), and the work done by gravity is \( W_g = mg \sin(34^\circ) \cdot d \). This calculates as: \[ \text{Net Work} = \frac{1}{2}mv^2 = mgd(\sin(34^\circ) - \mu_k \cos(34^\circ)) \] where \( d = 6.0 \) m. Solving for \( v \), we find \( v \approx 5.95 \text{ m/s} \).

Step 4: Calculate the Speed as the Snow Hits the Ground

For part (c), use the conservation of energy principle. As the snow chunk falls the height of 10.0 m, its potential energy is converted to kinetic energy. The total mechanical energy at the edge of the roof and on the ground is constant. Hence, \( \frac{1}{2}mv^2 + mgh_1 = \frac{1}{2}mv_f^2 \), where \( v \) is the speed calculated in step 3 and \( v_f \) is the final speed. Solving for \( v_f \), we find \( v_f \approx 15.77 \text{ m/s} \).

Key Concepts

Static Friction

Static friction is a force that keeps an object at rest when placed on a surface. It comes into play when there is a force trying to move the object, but the frictional force is strong enough to resist it. In this physics problem involving snow on a roof, static friction prevents the snow chunk from sliding down.

• For an object on a slope, static friction acts opposite to the direction of the gravitational force that pulls it downwards.
• The maximum static frictional force can be calculated using the formula: \( f_{static} = \mu_s N \), where \( \mu_s \) is the coefficient of static friction, and \( N \) is the normal force.
• In our scenario, the normal force equals the gravitational force component perpendicular to the roof, given by \( N = mg \cos(34^\circ) \).

The challenge is to find the minimum static friction coefficient \( \mu_s \) that prevents sliding. For this, it must balance the parallel component of gravity, leading us to \( \mu_s = \tan(34^\circ) \), which calculates to approximately 0.67. This means a coefficient of at least 0.67 is necessary to keep the snow static on the inclined roof.

Kinetic Friction

Once an object starts moving, static friction is no longer applicable, and kinetic friction takes over. Kinetic friction opposes the motion of a sliding object, albeit with a typically lower coefficient than static friction.

• Kinetic friction is described by the formula: \( f_{kinetic} = \mu_k N \), where \( \mu_k \) is the coefficient of kinetic friction, which in this case is 0.20.
• As the snow begins to slide, kinetic friction acts in the opposite direction of motion, reducing the net force and thus the net work done on the snow.

Calculating kinetic energy involves understanding that the net work done on an object is the change in its kinetic energy. Here, the work-energy principle helps us determine the snow's speed as it slips to the roof’s edge. The calculation takes into account both the work done by gravity and the work counteracted by kinetic friction.

Work-Energy Principle

The work-energy principle is a core concept in mechanics that relates the work done by all forces acting on an object to its change in kinetic energy. This principle is crucial for solving the snow chunk's journey down the roof.

• The total work done on the snow chunk is the difference between the work done by gravitational forces and the work done by friction.
• In the exercise, the formula is applied as:\[ \text{Net Work} = \frac{1}{2}mv^2 = mgd(\sin(34^\circ) - \mu_k \cos(34^\circ)) \]
• This setup allows us to solve for the snow's speed when falling off the roof. The calculated speed is approximately 5.95 m/s.

This illustrates how energy transformations inform about motion and provides insights into how mechanical work impacts kinetic energy. By understanding this principle, students can link force application directly to motion outcomes.

Conservation of Energy

The conservation of energy is a fundamental principle stating that energy in a closed system remains constant, transforming from one form to another without being created or destroyed.

• In the context of this problem, as the snow falls from the roof's edge to the ground, gravitational potential energy (related to height) gets converted entirely into kinetic energy (related to motion).
• The energy equation relates the kinetic energy at the edge of the roof to that at ground level:\[ \frac{1}{2}mv^2 + mgh_1 = \frac{1}{2}mv_f^2 \]
• Here, \( v \) is the initial speed when sliding off the roof, and \( v_f \) is the final speed upon impact with the ground, which results in a speed of roughly 15.77 m/s.

By grasping energy conservation, one better understands how potential energy changes into kinetic energy, exemplifying a smooth transition of energy states as the snow chunk descends.