Question

A 28-kg rock approaches the foot of a hill with a speed of 15 m/s. This hill slopes upward at a constant angle of 40.0\(^\circ\) above the horizontal. The coefficients of static and kinetic friction between the hill and the rock are 0.75 and 0.20, respectively. (a) Use energy conservation to find the maximum height above the foot of the hill reached by the rock. (b) Will the rock remain at rest at its highest point, or will it slide back down the hill? (c) If the rock does slide back down, find its speed when it returns to the bottom of the hill.

Short answer

(a) 7.4 m, (b) The rock slides back, (c) 6.1 m/s.

Step-by-step solution

Identify Energy Conservation Components

The mechanical energy of the rock is conserved, consisting of kinetic energy (KE) and gravitational potential energy (PE). At the base, the total energy is all kinetic. At the maximum height, it is all potential.

Calculate Initial Kinetic Energy

Initial kinetic energy at the base of the hill is calculated using the formula: \( KE_i = \frac{1}{2}mv^2 = \frac{1}{2} \times 28 \times 15^2 = 3150 \, \text{J} \).

Calculate Work Done Against Friction

Calculate the force of kinetic friction: \( f_k = \mu_k mg \cos \theta = 0.20 \times 28 \times 9.8 \times \cos(40^\circ) = 42.23 \, \text{N} \).The work done against friction as the rock moves distance \(d\) up the hill is \( W_f = f_k \cdot d \).

Calculate Work Done Against Gravity

The work done against gravity is given by the change in potential energy: \( PE = mgh \). Since energy is conserved: \( KE_i = mgh + W_f \).

Express Height in Terms of Distance

The height \(h\) is related to distance \(d\) by \( h = d \sin \theta \).Substitute into the energy equation: \( KE_i = mgd \sin \theta + f_k \cdot d \).

Solve for Distance \(d\)

Rearrange this equation to solve for \(d\): \( 3150 = 28 \cdot 9.8 \cdot d \sin 40^\circ + 42.23d \).This simplifies to \( d = \frac{3150}{274.19} \approx 11.5 \text{ m} \).

Calculate Maximum Height \(h\)

Substitute \(d\) back into the height equation: \( h = 11.5 \sin(40^\circ) \approx 7.4 \, \text{m} \).

Determine if Rock Slides Back

Check if the static friction is enough to hold the rock at the top. The maximum static friction is \( f_s = \mu_s mg \cos \theta = 0.75 \times 28 \times 9.8 \times \cos(40^\circ) \approx 158 \, \text{N} \).If the gravitational component \( mg \sin \theta = 28 \times 9.8 \times \sin(40^\circ) \approx 176.3 \, \text{N} \) exceeds \(f_s\), the rock slides back.

Calculate Speed at Bottom

If the rock slides back, use energy conservation again: \( mgh = \frac{1}{2}mv^2 + W_f \text{ for descent} \). Rearrange to solve for final speed \( v \): \( v = \sqrt{2gh - \frac{2W_f}{m}} \approx \sqrt{2 \times 9.8 \times 7.4 - \frac{2 \times 42.23 \times 11.5}{28}} = 6.1 \text{ m/s} \).

Key concepts

Kinetic Energy

Kinetic energy is all about the motion of an object. Every moving object possesses kinetic energy, which depends on two main factors: its mass and its velocity. You can calculate the kinetic energy (KE) of an object using the formula:\[KE = \frac{1}{2} m v^2\]where \(m\) is the mass in kilograms and \(v\) is the velocity in meters per second.
In our exercise, the rock initially has all its energy in the form of kinetic energy as it approaches the hill at a speed of 15 m/s. The kinetic energy at this point is calculated to be 3150 Joules. This energy will be converted as the rock starts its ascend.

• Kinetic energy increases with the square of the velocity. So doubling the speed results in four times the kinetic energy.
• As the rock moves uphill, part of its kinetic energy is converted into other forms such as potential energy and lost due to friction.

Gravitational Potential Energy

Gravitational potential energy is the energy stored due to an object's position above the ground. The higher an object is, the more potential energy it has.
This energy can be calculated using the formula:\[PE = mgh\]where \(m\) is the mass, \(g\) is the acceleration due to gravity (approximately 9.8 m/s²), and \(h\) is the height above the reference point.
As the rock climbs the hill, its kinetic energy is transferred into gravitational potential energy.
At its maximum height of 7.4 meters, all the surviving energy is now potential. This transformation is a classic example of energy conservation in action.

• Potential energy depends linearly on height. So, if the height doubles, the potential energy also doubles.
• In the absence of friction, all the initial kinetic energy would be converted to potential energy at the highest point.

Friction in Physics

Friction is a force that opposes motion between two surfaces in contact. Two types relevant to our rock scenario are kinetic friction and static friction.

• Kinetic Friction: This occurs when two objects are moving against each other. It acts to slow down the rock as it goes upward. In our example, the kinetic friction was calculated as 42.23 N. The work done against this friction is subtracted from the rock's mechanical energy.
• Static Friction: This prevents motion when at least one object is at rest. At the highest point, static friction tries to keep the rock from sliding back down. However, if the gravitational pull is greater than the maximum static friction force (158 N), the rock will start moving again.

Friction converts mechanical energy into thermal energy, which is why not all the kinetic energy converts to potential energy in real scenarios. Understanding friction helps you realize why objects don't keep moving indefinitely or can stay put on inclines without sliding back.