Problem 73 - University Physics with Modern Physics

Question

A wooden block with mass 1.50 kg is placed against a compressed spring at the bottom of an incline of slope 30.0\(^\circ\) (point \(A\)). When the spring is released, it projects the block up the incline. At point \(B\), a distance of 6.00 m up the incline from A, the block is moving up the incline at 7.00 m/s and is no longer in contact with the spring. The coefficient of kinetic friction between the block and the incline is \(\mu_k =\) 0.50. The mass of the spring is negligible. Calculate the amount of potential energy that was initially stored in the spring.

Step-by-Step Solution

Answer

The initial potential energy stored in the spring is approximately 119 J.

Step 1: Identify Known Quantities

We start by identifying the known variables in the problem. These are:- Mass of the block, \(m = 1.50\, \text{kg}\)- Slope of the incline, \( \theta = 30.0^\circ \)- Distance traveled by the block, \(s = 6.00\, \text{m}\)- Final speed at point \(B, v_B = 7.00\, \text{m/s}\)- Coefficient of kinetic friction, \(\mu_k = 0.50\)We need to calculate the initial potential energy stored in the spring, \(PE_s\).

Step 2: Calculate the Work Done Against Friction

The work done against friction, \(W_f\), can be computed using the formula:\[W_f = \mu_k \cdot m \cdot g \cdot \cos(\theta) \cdot s\]Where \(g = 9.81\, \text{m/s}^2\). Plugging in the values gives:\[W_f = 0.50 \cdot 1.50 \cdot 9.81 \cdot \cos(30^\circ) \cdot 6.00\]By calculating:\[W_f = 0.50 \cdot 1.50 \cdot 9.81 \cdot 0.866 \cdot 6.00\ = 38.21\, \text{J}\]

Step 3: Calculate the Change in Kinetic Energy

The kinetic energy at point \(A\) is 0 since the block starts from rest. The change in kinetic energy as the block moves to \(B\) is:\[\Delta KE = \frac{1}{2} m v_B^2 - 0 = \frac{1}{2} \cdot 1.50 \cdot (7.00)^2\]Calculating yields:\[\Delta KE = 0.75 \cdot 49 = 36.75\, \text{J}\]

Step 4: Calculate the Change in Gravitational Potential Energy

The change in gravitational potential energy (GPE) is given by:\[\Delta PE_g = m \cdot g \cdot h\]Where \(h = s \cdot \sin(\theta)\). Thus:\[h = 6.00 \cdot \sin(30^\circ) = 3.00\, \text{m}\]Then:\[\Delta PE_g = 1.50 \cdot 9.81 \cdot 3.00 = 44.145\, \text{J}\]

Step 5: Calculate the Total Energy and Initial Spring Potential Energy

The total mechanical energy change due to motion from point \(A\) to point \(B\) must be equal to the initial potential energy stored in the spring. This total comprises the work done against friction, kinetic energy, and gravitational potential energy changes:\[PE_s = W_f + \Delta KE + \Delta PE_g\]Thus:\[PE_s = 38.21 + 36.75 + 44.145 = 119.105\, \text{J}\]

Step 6: Round the Final Answer

Round \(PE_s\) to three significant figures, as the given values have three significant figures. Hence, the initial potential energy stored in the spring is:\[PE_s = 119\, \text{J}\]

Key Concepts

Potential Energy

Potential energy is the energy an object has because of its position or arrangement. In the context of the exercise, we are dealing with the potential energy stored in a compressed spring. When a spring is compressed, it holds energy that can be released to do work, such as accelerating a block up an inclined plane. The initial potential energy stored in the spring is calculated based on how much work the spring can do, which helps us to understand how the block is moved along the incline. By knowing the potential energy, we can predict the motion of the block once the spring is released, as this energy will be converted into other forms of energy like kinetic energy.

Kinetic Friction

Kinetic friction occurs between moving surfaces and acts against the motion of an object. It is the force that needs to be overcome for an object to continue moving. In the exercise, the wooden block slides up the inclined plane, experiencing kinetic friction that opposes its movement. The frictional force can be calculated using the formula:

• The force of friction, \(F_f = \mu_k \cdot m \cdot g \cdot \cos(\theta)\), where \(\mu_k\) is the coefficient of kinetic friction, \(m\) is mass, \(g\) is gravitational acceleration, and \(\theta\) is the incline angle.

Kinetic friction converts some of the block's mechanical energy into thermal energy, reducing the speed at which the block travels upward. Understanding kinetic friction is essential for calculating how much extra energy is required to overcome this loss.

Inclined Plane

An inclined plane is a flat, sloped surface that helps to lift or lower loads, making work easier by reducing the force needed. In physics, it is often used in problems to illustrate the conversion of energy types and the forces at play. The motion of our wooden block up an incline involves gravitational effects and friction. Gravitational force pulls the block down the plane depending on its angle, typically making the ascent more demanding. Inclined planes are efficient teaching tools for energy conversion, as they involve angles and height, which allows us to explore potential and kinetic energy in a straightforward manner. For our example, understanding the steepness of the plane (30 degrees) helps determine how much gravitational potential energy changes as the block moves.

Energy Conservation

The principle of energy conservation states that energy in a closed system remains constant but can change forms. In our scenario, energy is transformed from potential energy in the spring to kinetic energy in the moving block, and eventually to gravitational potential energy as the block ascends.

• Initially, energy is stored in the compressed spring.
• As the spring releases, this energy converts into the block's kinetic energy (movement energy).
• Friction along the incline converts part of this energy into heat.
• The block also gains gravitational potential energy as it rises on the incline.

By conserving energy from start to finish, we can calculate how much initial potential energy was needed to overcome friction, move the block, and lift it to a particular height, demonstrating the effectiveness of energy conversion principles.