Problem 66 - University Physics with Modern Physics

Question

A 5.00-kg package slides 2.80 m down a long ramp that is inclined at 24.0\(^\circ\) below the horizontal. The coefficient of kinetic friction between the package and the ramp is \(\mu_k\) \(=\) 0.310. Calculate (a) the work done on the package by friction; (b) the work done on the package by gravity; (c) the work done on the package by the normal force; (d) the total work done on the package. (e) If the package has a speed of 2.20 m/s at the top of the ramp, what is its speed after it has slid 2.80 m down the ramp?

Step-by-Step Solution

Answer

(a) -38.92 J, (b) 55.92 J, (c) 0 J, (d) 17.0 J, (e) 3.41 m/s.

Step 1: Calculate the Force of Friction

First, determine the force of friction. The force of friction can be calculated using the formula \( f_k = \mu_k \cdot N \), where \( N \) is the normal force. On an inclined plane, \( N = mg\cos\theta \), where \( m = 5.00 \text{ kg} \), \( g = 9.81 \text{ m/s}^2 \), and \( \theta = 24.0^\circ \). Calculate: \[ N = 5.00 \cdot 9.81 \cdot \cos(24.0^\circ) \approx 44.85 \text{ N} \]\[ f_k = 0.310 \times 44.85 \approx 13.90 \text{ N} \]

Step 2: Calculate Work Done by Friction

The work done by friction is given by \( W_f = f_k \cdot d \cdot \cos(180^\circ) \), where \( d = 2.80 \text{ m} \) is the distance. \[ W_f = 13.90 \times 2.80 \times (-1) \]\[ W_f = -38.92 \text{ J} \]

Step 3: Calculate Work Done by Gravity

The work done by gravity is given by \( W_g = mgd\sin\theta \). Use the values \( m = 5.00 \text{ kg}, g = 9.81 \text{ m/s}^2, \theta = 24.0^\circ, d = 2.80 \text{ m} \).\[ W_g = 5.00 \times 9.81 \times 2.80 \times \sin(24.0^\circ) \approx 55.92 \text{ J} \]

Step 4: Calculate Work Done by Normal Force

The work done by the normal force is zero because the normal force acts perpendicular to the direction of motion. Hence,\[ W_n = 0 \text{ J} \]

Step 5: Calculate Total Work Done

The total work done on the package is the sum of the work done by friction, gravity, and the normal force.\[ W_{total} = W_f + W_g + W_n \]\[ W_{total} = -38.92 + 55.92 + 0 = 17.0 \text{ J} \]

Step 6: Calculate Final Speed

Use the work-energy principle to find the final speed of the package. The net work done on the package is equal to the change in kinetic energy.\[ W_{total} = \frac{1}{2} mv_f^2 - \frac{1}{2} mv_i^2 \]Rearrange to find \( v_f \):\[ 17.0 = \frac{1}{2} \cdot 5.00 \cdot v_f^2 - \frac{1}{2} \cdot 5.00 \cdot 2.20^2 \]Solve for \( v_f \):\[ 17.0 = \frac{1}{2} \cdot 5.00 \cdot (v_f^2 - 4.84) \]\[ 17.0 = 2.50(v_f^2 - 4.84) \]\[ 6.8 = v_f^2 - 4.84 \]\[ v_f^2 = 11.64 \]\[ v_f = \sqrt{11.64} \approx 3.41 \text{ m/s} \]

Key Concepts

Kinetic Friction

Kinetic friction is a force that opposes the motion of objects sliding against each other. It plays a crucial role in slowing things down. When an object moves across a surface, kinetic friction acts in the opposite direction of the object's movement.
In our exercise, the coefficient of kinetic friction (\( \mu_k = 0.310 \)) tells us how "sticky" the two surfaces are. The normal force (\( N \)) presses the object against the surface. It depends on the object's weight and the angle of the slope.To calculate the force of kinetic friction, we use the equation:

• \( f_k = \mu_k \times N \)

This frictional force results in energy loss, affecting how objects move over time.
In scenarios like our inclined plane, understanding kinetic friction is essential to predict the motion and final speed of the object.

Inclined Plane

An inclined plane is a flat surface tilted at an angle, and it's a simple machine used to make lifting objects easier. In physics problems, an inclined plane helps demonstrate how gravity, friction, and other forces interact.
When an object is on an inclined plane, the gravitational force can be split into two components:

• One parallel to the plane, causing the object to slide down.
• Another perpendicular to the plane, acting as the normal force.

For our 24.0-degree incline, the parallel component is calculated using \( mg\sin(\theta) \) and the perpendicular using \( mg\cos(\theta) \).These components allow us to determine how the object moves down the plane and how friction affects that motion. Analyzing such setups helps us study complex real-world interactions in a simpler manner.

Net Work

In physics, work is done when a force moves an object over a distance. Net work is the total work done by all forces on an object. It's vital in understanding how energy transfers between forms and how it affects an object's motion.
When calculating net work, consider contributions from all forces:

• Work done by gravity.
• Work done by friction (often negative since it opposes motion).
• Work by normal force (usually zero as it's perpendicular to movement).

In our scenario, the net work done is:\( W_{total} = W_f + W_g + W_n \)This net work translates into a change in kinetic energy, as described by the work-energy principle. This principle is key to predicting how the object's speed changes after moving a certain distance. By adding or subtracting energy through work, we can determine the final velocity of the package sliding down the ramp.