3. A spring of constant k, compressed a distance x, is used to launch a mass m up a frictionless slope at angle 0. Find an expression for the maximum distance along the slope that the mass moves after leaving the spring.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
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![**Problem 3: Analysis of a Spring-Launched Mass on a Frictionless Incline**
A spring with a constant \( k \) is compressed by a distance \( x \) and is used to propel a mass \( m \) up an incline that is completely frictionless and angled at \( \theta \). Determine an expression for the maximum distance the mass will travel along the incline after it detaches from the spring.
**Explanation:**
To solve this problem, we need to analyze the energy transformations involved:
1. **Initial Energy**: The initial energy comes from the potential energy stored in the compressed spring, given by \( \frac{1}{2} k x^2 \).
2. **Final Energy**: As the mass moves up the incline, the spring’s potential energy transforms into gravitational potential energy. At the maximum distance, the kinetic energy will be zero because the mass momentarily stops.
3. **Gravitational Potential Energy Increase**: As the mass moves up, it's potential energy changes by \( mgh \), where \( h \) can be expressed in terms of the distance up the incline \( d \), using \( h = d \sin \theta \).
**Energy Conservation Equation**:
\[ \frac{1}{2} k x^2 = mg(d \sin \theta) \]
**Solving for \( d \)**:
\[ d = \frac{\frac{1}{2} k x^2}{mg \sin \theta} \]
The above equation provides the expression for the maximum distance \( d \) the mass travels along the incline.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4143f45a-5c96-4449-af2f-abda697f3e91%2Fbe8f739e-4207-402a-b2ff-9240d2b686d2%2Fw8z2ix_processed.jpeg&w=3840&q=75)
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