Spring-Mass System and Combination of Springs

Spring-Mass System and Combination of Springs

A spring-mass system has \( m = 2.5 \, \text{kg}, k = 1000 \, \text{N/m} \). What is its angular frequency?

**Mass-spring dynamics** show T depends on mass and stiffness, independent of amplitude for ideal spring. Given T and m, k = 4π² m/T² extracted, and energy E = ½ k A² connects amplitude to total mechanical energy, illustrating isochronism. ω = √((k/m)) = √((1000/2.5)) = √(400) = 20 rad/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 20 rad/s follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Spring-Mass System and Combination of Springs

A spring-mass system has \( m = 0.8 \, \text{kg}, k = 320 \, \text{N/m} \). If displaced by \( 5 \, \text{cm} \), what i

**Mass-spring dynamics** show T depends on mass and stiffness, independent of amplitude for ideal spring. Given T and m, k = 4π² m/T² extracted, and energy E = ½ k A² connects amplitude to total mechanical energy, illustrating isochronism. Total energy: E = (1/2) k A² . A = 0.05 m, k = 320 N/m . E = 0.5 × 320 × (0.05)² = 0.5 × 320 × 0.0025 = 0.4 J . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.4 J follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Spring-Mass System and Combination of Springs

A particle’s motion is given by \( x = 2 \cos (5t - \frac{\pi}{4}) \) (in m). What is its kinetic energy at \( x = 1 \,

**Oscillation of mass attached to spring** follows Hooke's law F = -k x, SHM with ω = √(k/m). Effective stiffness for two identical springs in parallel doubles, k_eff = 2k, increasing frequency by √2, while series halves stiffness to k/2, lowering frequency. Energy E = ½ k_eff A². Total energy: E = (1/2) k A² = (1/2) m ω² A² = 0.5 × 1 × 5² × 2² = 50 J . Potential energy: U = (1/2) m ω² x² = 0.5 × 1 × 25 × 1² = 12.5 J . Kinetic energy: K = E - U = 50 - 12.5 = 37.5

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Spring-Mass System and Combination of Springs

A spring-mass system has \( m = 0.6 \, \text{kg}, k = 240 \, \text{N/m} \). If displaced by \( 7 \, \text{cm} \), what i

**Spring-mass system** has period T = 2π√(m/k), frequency f = (1/2π)√(k/m), ω = √(k/m), where k spring constant (N/m) and m mass (kg). For parallel combination, k_eff = k₁ + k₂, series gives 1/k_eff = 1/k₁ + 1/k₂, affecting ω = √(k_eff/m) and T = 2π√(m/k_eff). Total energy: E = (1/2) k A² . A = 0.07 m, k = 240 N/m . E = 0.5 × 240 × (0.07)² = 0.5 × 240 × 0.0049 = 0.588 J . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.588 J

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Spring-Mass System and Combination of Springs

A spring-mass system has \( m = 0.25 \, \text{kg}, k = 100 \, \text{N/m} \). If displaced by \( 6 \, \text{cm} \), what

**Mass-spring dynamics** show T depends on mass and stiffness, independent of amplitude for ideal spring. Given T and m, k = 4π² m/T² extracted, and energy E = ½ k A² connects amplitude to total mechanical energy, illustrating isochronism. Total energy: E = (1/2) k A² . A = 0.06 m, k = 100 N/m . E = 0.5 × 100 × (0.06)² = 0.5 × 100 × 0.0036 = 0.18 J . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.18 J follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Spring-Mass System and Combination of Springs

What effect does tripling the spring constant have on the period of a spring-mass system if the mass is also tripled?

**Oscillation of mass attached to spring** follows Hooke's law F = -k x, SHM with ω = √(k/m). Effective stiffness for two identical springs in parallel doubles, k_eff = 2k, increasing frequency by √2, while series halves stiffness to k/2, lowering frequency. Energy E = ½ k_eff A². Period T = 2π √((m/k)) . If k' = 3k and m' = 3m , then T' = 2π √((3m/3k)) = 2π √((m/k)) = T , so the period remains unchanged. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result It remains unchanged

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Spring-Mass System and Combination of Springs

A spring of \( k = 400 \, \text{N/m} \) is attached to a \( 1 \, \text{kg} \) mass. What is the angular frequency?

**Oscillation of mass attached to spring** follows Hooke's law F = -k x, SHM with ω = √(k/m). Effective stiffness for two identical springs in parallel doubles, k_eff = 2k, increasing frequency by √2, while series halves stiffness to k/2, lowering frequency. Energy E = ½ k_eff A². ω = √((k/m)) = √((400/1)) = 20 rad/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 20 rad/s follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Spring-Mass System and Combination of Springs

A spring-mass system has \( m = 4 \, \text{kg}, k = 400 \, \text{N/m} \). If displaced by \( 20 \, \text{cm} \), what is

**Oscillation of mass attached to spring** follows Hooke's law F = -k x, SHM with ω = √(k/m). Effective stiffness for two identical springs in parallel doubles, k_eff = 2k, increasing frequency by √2, while series halves stiffness to k/2, lowering frequency. Energy E = ½ k_eff A². Potential energy: U = (1/2) k x² . At x = 10 cm = 0.1 m : U = (1/2) × 400 × (0.1)² = 0.5 × 400 × 0.01 = 2 J . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result

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Spring-Mass System and Combination of Springs

A spring-mass system has \( m = 0.4 \, \text{kg}, k = 160 \, \text{N/m} \). If displaced by \( 6 \, \text{cm} \), what i

**Spring-mass system** has period T = 2π√(m/k), frequency f = (1/2π)√(k/m), ω = √(k/m), where k spring constant (N/m) and m mass (kg). For parallel combination, k_eff = k₁ + k₂, series gives 1/k_eff = 1/k₁ + 1/k₂, affecting ω = √(k_eff/m) and T = 2π√(m/k_eff). Total energy: E = (1/2) k A² . A = 0.06 m, k = 160 N/m . E = 0.5 × 160 × (0.06)² = 0.5 × 160 × 0.0036 = 0.288 J . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.288 J

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Spring-Mass System and Combination of Springs

A spring-mass system oscillates with \( T = 0.7 \, \text{s} \) when \( m = 0.7 \, \text{kg} \). What is the spring const

**Oscillation of mass attached to spring** follows Hooke's law F = -k x, SHM with ω = √(k/m). Effective stiffness for two identical springs in parallel doubles, k_eff = 2k, increasing frequency by √2, while series halves stiffness to k/2, lowering frequency. Energy E = ½ k_eff A². T = 2π √((m/k)) . 0.7 = 2π √((0.7/k)) ⇒ (0.7/2π) = √((0.7/k)) . (0.1114)² = (0.7/k) ⇒ k = (0.7/0.01241) ≈ 56.4 N/m . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 56.4 N/m follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Spring-Mass System and Combination of Springs

A spring-mass system has \( m = 0.9 \, \text{kg}, k = 360 \, \text{N/m} \). If displaced by \( 4 \, \text{cm} \), what i

**Mass-spring dynamics** show T depends on mass and stiffness, independent of amplitude for ideal spring. Given T and m, k = 4π² m/T² extracted, and energy E = ½ k A² connects amplitude to total mechanical energy, illustrating isochronism. Total energy: E = (1/2) k A² . A = 0.04 m, k = 360 N/m . E = 0.5 × 360 × (0.04)² = 0.5 × 360 × 0.0016 = 0.288 J . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.288 J follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Spring-Mass System and Combination of Springs

Two identical springs (\( k = 75 \, \text{N/m} \)) are attached to a \( 1.5 \, \text{kg} \) mass as in Fig. 13.14. What

**Oscillation of mass attached to spring** follows Hooke's law F = -k x, SHM with ω = √(k/m). Effective stiffness for two identical springs in parallel doubles, k_eff = 2k, increasing frequency by √2, while series halves stiffness to k/2, lowering frequency. Energy E = ½ k_eff A². Effective kₑff = 2k = 2 × 75 = 150 N/m . ω = √((kₑff/m)) = √((150/1.5)) = √(100) = 10 rad/s . v = (ω/2π) = (10/2 × 3.14) ≈ 1.59 Hz . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result

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